Afzal Khan said he could not hold his mum's hand as she lay dying, and lost his father and mother-in-law within days of one another.
"Grieving families like mine deserved to understand what happened," he said.
UK PM Boris Johnson said he was "of course committed" to holding an inquiry "as soon as it's right to do so".
Speaking at Prime Minister's Questions, Mr Khan, MP for Manchester, Gorton, said: "Grieving families like mine want, and deserve, to understand what happened and if anything could have been done to prevent this tragedy.
"So will the prime minister today commit to launching a full public inquiry into the handling of the pandemic as soon as current restrictions are lifted?"
Mr Johnson replied: "As soon as it would not be an irresponsible diversion of the energies of the key officials involved, we, of course, are committed to an inquiry to learn the lessons to make sure that something like this can never happen again."
The matter was also raised during Prime Minister's Questions by Labour leader Sir Keir Starmer.
He described the 126,000 deaths linked to Covid-19 as "shocking", adding: "Behind every one of those numbers is a grieving family.
"As soon as restrictions lift, there must be a full public inquiry because that is the only way we can get to the bottom of the many mistakes made during the pandemic and find justice for those who have suffered so much."
In July, the prime minister promised an "independent inquiry"; last week Housing Secretary Robert Jenrick said now was "not the time".
He told Radio 4's Today programme there would come a time when the government could do something "more formal, look back and learn the lessons" but the "sole focus" now had to be on the vaccine rollout and the economic recovery.
Deputy chief medical officer for England, Professor Jonathan Van-Tam, also said last week that an inquiry now would be "an extra burden that wasn't necessary".
If the grass is greener on the other side, you can bet the water bill is higher.