It was a clarion call to chief executives everywhere.

In 2018, Laurence D. Fink, the longtime chief executive of BlackRock, the world’s largest asset manager, urged corporate leaders to assess the societal impact of their businesses, embrace diversity and consider how climate change could affect long-term growth.

“Companies,” Mr. Fink wrote in his annual letter to chief executives, “must ask themselves: What role do we play in the community? How are we managing our impact on the environment? Are we working to create a diverse work force? Are we adapting to technological change?”

Nearly five years on, those words have put BlackRock on the back foot amid the increasingly acrimonious and politicized debate over investing with environmental, social and governance — or E.S.G. — goals in mind. Republicans are accusing the firm of “woke capitalism.” Progressives are calling BlackRock out for “greenwashing,” saying its message to companies doesn’t go far enough.

In recent months, more than half a dozen Republican state treasurers and comptrollers have stepped up their attacks on BlackRock, which manages $8 trillion in assets and invests on behalf of hundreds of public pensions. On Dec. 1, Florida’s chief financial officer said the state was pulling $2 billion from BlackRock because it was “undemocratic” for a big asset manager to try to change society. Eight days later, North Carolina’s treasurer called for Mr. Fink to resign because he had pushed corporations to cut carbon emissions.

At the same time, progressive critics are asking whether the E.S.G. mutual funds and exchange-traded funds pushed by BlackRock and other asset managers are any different from decades-old investment products given a green makeover. In September, New York City’s comptroller, Brad Lander, a Democrat, sent Mr. Fink a letter expressing concern that BlackRock was backsliding on its commitment to promote net-zero emission standards.

“Knowing what Larry knows now, I suspect there are elements of his C.E.O. letters that he would have omitted or written differently,” said Terrence Keeley, BlackRock’s former global head of the official institutions group. “He took some big risks in his C.E.O. letters, and that has led to some of the bitter fruit he’s harvesting now.” Mr. Keeley, who retired from BlackRock this year, oversaw sovereign wealth funds, pensions and central banks.

Investing with consideration for climate change, diversity, gender and pay equity, the welfare of employees, and the impact of technology on society — broadly lumped together under the E.S.G. banner — has become a big focus of asset managers and companies in recent years, with BlackRock leading the charge. Some on Wall Street and in corporate America see a clear benefit in embracing the approach, given the growing consumer focus on sustainability.

But a big challenge is that, in the absence of regulatory guidance, what constitutes E.S.G. investing often lies in the eye of the beholder. A company incorporating elements of the trend is ripe for attack from politicians and activists for doing too much or too little.

Recently, Bluebell Capital, a small hedge fund in London, called for Mr. Fink’s ouster, accusing him of flip-flopping on his support for reducing emissions even as he “succeeded in the remarkable task of alienating” parties on both sides of the E.S.G. debate.

Mr. Fink, who declined to be interviewed for this article, followed up his 2018 letter, titled “A Sense of Purpose,” a year later by writing that “environmental, social and governance issues will be increasingly material to corporate valuations.” He signaled to investors that BlackRock would play a leading role in promoting sustainable investment products and using its proxy power — or power to vote on behalf of those whose assets the firm manages — to push companies toward adopting carbon emission reduction plans.

BlackRock quickly became a leader in the United States in E.S.G. investing, churning out mutual funds and exchange-traded funds billed as products that allowed investors to put their money into companies that supported climate change initiatives, promoted diversity in the workplace and avoided countries where workers lack basic protections.

“When Larry really embraced E.S.G., it became a big thing, and everyone was really energized,” said Peter McKillop, a former corporate spokesman for BlackRock who now runs a newsletter and website focused on climate change. At the time, neither Mr. Fink nor BlackRock’s leadership considered the potential for backlash, Mr. McKillop said. “It wasn’t really thought out.”

BlackRock acknowledged the issue. “Many people have opinions on how our clients’ assets should be invested,” a spokesman said in an emailed statement. “However, our fiduciary duty is to each of our clients. The money we manage belongs to them — not to politicians, activists, NGOs or commentators.”

The asset manager has increased its messaging along those lines recently, unrolling an advertising campaign meant to demystify its business. In one 30-second TV spot that aired in September, the narrator conveys that “from the plains to the coasts,” BlackRock is in the business of helping Americans “invest for their future and help communities thrive.”

In the past year, Mr. Fink has sought to address the criticism by saying BlackRock is not ideologically driven. In his letter to chief executives this year, he wrote that the firm had no plans to divest from fossil fuel investments and wasn’t pressuring any clients to do so.

At a conference sponsored by DealBook and The New York Times last month, Mr. Fink said, “I actually believe we’re going to need hydrocarbons for 70 years.”

So far, Republican state politicians have pulled a little over $4 billion from BlackRock — a pittance compared with the $133 billion that the firm has taken in this year from U.S. investors. Still, the calls from Republican treasurers to withdraw state money from BlackRock over its E.S.G. policies is accelerating.

In addition to Florida and North Carolina, state officials from Arkansas, Arizona, Louisiana, Missouri and South Carolina have withdrawn money from BlackRock. Utah and West Virginia have announced plans to do so.

Mr. McKillop said he believed that the criticism from Republican officials had particularly gotten to Mr. Fink, a Democrat, which is why he has been emphasizing that BlackRock has substantial oil and gas investments.

“He doesn’t want to lose money even if it’s a de minimis amount,” Mr. McKillop said.

BlackRock is not the only big asset manager to come under fire in the United States.

In mid-December, representatives from State Street appeared alongside a BlackRock executive at a Texas legislative hearing about the impact of E.S.G. investing on the state’s fossil fuel companies. At the hearing, Dalia Blass, a BlackRock executive, pointed out that the firm had invested $107 billion in public Texas energy companies on behalf of its clients, and had generated above-average returns for Texas pension clients.

At the same time, the Republican staff of the Senate Banking Committee recently put out a report criticizing BlackRock, Vanguard and State Street for using their investing muscle to push for corporate proxy votes for measures advocated by progressives.

“Each of these firms proudly uses the voting power gained from their investors’ money to advance liberal social goals,” the report said.

BlackRock and State Street said they disagreed with the findings. BlackRock said the report was built on “flawed premises” and risked “harming millions of everyday investors that rely on mutual funds and exchange-traded funds.” State Street said the report ignored the “critical role of index funds in helping average Americans save for retirement by providing access to low-cost investments.”

Vanguard, one of the biggest marketers of index funds, said its mission was to empower “everyday investors to reach their long-term financial goals.”

Several of the state Republican leaders who have come out accusing BlackRock of engaging in “woke capitalism” are members of the State Financial Officers Foundation, whose website prominently displays an emblem that says: “Educating Americans on the Dangers of ESG.”

Jimmy Patronis, the chief financial officer of Florida, is a foundation member. In announcing his decision to pull $2 billion from BlackRock, Mr. Patronis said he did not support the firm’s “social engineering.”

In an interview, Mr. Patronis said his decision was largely a financial one and based on BlackRock's “middle of the pack” performance, although he added that Mr. Fink’s “agenda has just given us some room for concern.”

Some progressive activists and Democratic politicians contend that BlackRock, in trying to appease conservative critics, is backtracking from climate change commitments. Others have claimed that investment products focused on E.S.G. are not as transformational as billed. In October, climate activists dumped a bucket of coal inside BlackRock’s Manhattan headquarters, saying it was not doing enough on climate change.

Tariq Fancy, a former head of sustainable investing at BlackRock, said some progressives were starting to find Mr. Fink’s support for E.S.G. somewhat hollow. “E.S.G., to some degree, is a smoke screen,” Mr. Fancy said.

One of the more outspoken Democratic critics of BlackRock is Mr. Lander, the New York City comptroller, whose office has $43 billion in public pension funds invested with the asset manager. In 2020, BlackRock worked with New York to rid $3 billion worth of fossil fuel investments from two city employee pensions representing 700,000 people.

In his September letter, Mr. Lander urged Mr. Fink not to waver from his commitment to push companies toward a net-zero carbon emissions standard and chastised BlackRock for voting against some shareholder resolutions that asked “banks and insurers to stop financing new fossil fuel projects.”

BlackRock, in a Nov. 2 response to Mr. Lander, said, “Our role is not to engineer a specific decarbonization outcome in the real economy.”

Shaquana Chaneyfield, a spokeswoman for Mr. Lander, said the response had disappointed the comptroller. “We can only conclude that they are not serious about aligning their climate rhetoric with their actions,” she said.

Hans Taparia, a clinical associate professor at the New York University Stern School of Business, called E.S.G. a marketing strategy aimed at investors who wanted to feel that they were making a difference with their money.

“Real change with respect to E.S.G. would mean a drop in profits for many corporations, which is why we are witnessing something between infinitesimal change and greenwashing,” Mr. Taparia said.