Mr Rooney's wife Coleen is being sued for libel by Rebekah Vardy, who denies Mrs Rooney's accusation that she leaked private information about her.
On Tuesday, Mr Rooney said Mr Hodgson told him the Football Association (FA) wanted to avoid "problems and distractions" at Euro 2016.
Mr Vardy attended court on day six.
"They asked me, as captain, would I be able to speak to Mr Vardy on issues regarding his wife and I think we all knew that it was an awkward subject," said Mr Rooney, referring to Mr Hodgson and his then assistant Gary Neville.
"I'd need to speak to Mr Vardy and ask him to speak to his wife and ask him to ask his wife to calm down."
"Ask his wife to calm down?" queried Mrs Vardy's barrister Hugh Tomlinson. "She wasn't dancing on tables..."
"No, she wasn't, as far as I'm aware," Mr Rooney replied. But, he said, there had been negative media coverage relating to Mrs Vardy's Euro 2016 column in the Sun.
England's all-time record goalscorer said he "carried out that instruction" to have a word with his teammate and that it was Mr Vardy's business if the message was relayed to Mrs Vardy or not.
He said it "wasn't my place to speak to Mrs Vardy" directly but that he had "100%" passed the message on to his teammate.
"It was an awkward situation for me and I'm sure it was an awkward situation for Mr Vardy, but I felt it was in the best interests of the team," Mr Rooney said.
"As England captain I would always try and protect the players in public as much as I could," he continued.
Mr Rooney said Mrs Vardy was "almost there with the team" during the tournament in France.
He alleged that she had been calling her husband on FaceTime while the players were hanging out together during their downtime.
Former England men's football team captain, Mr Rooney, added that Mrs Vardy's husband, the Leicester City striker, was not someone "I have ever particularly been friends with on a social level".
Derby County manager Mr Rooney, 36, wrote last year in The Times that despite their wives' dispute, he would have tried to lure Mr Vardy, 35, out of international retirement for Euro 2020 had he been in charge.
In a statement issued by Mr Vardy's representatives to the PA news agency outside the hearing, the former England striker denied that Mr Rooney had spoken to him about getting Mrs Vardy to "calm down".
"Wayne is talking nonsense," Mr Vardy said. "He must be confused because he never spoke to me about issues concerning Becky's [Mrs Vardy's] media work at Euro 2016.
"There was nothing to speak about, I know this because I discuss everything with Becky."
However, Mr Vardy's statement will not have any impact on the trial as he is not due to be called as a witness.
Mrs Vardy's libel action was sparked by a viral social media post from October 2019, in which Mrs Rooney said she had carried out a sting operation to find out who had been passing information about her life, taken from her private Instagram account, to the Sun newspaper.
She said the fake stories she had posted on her Instagram stories in an effort to find the perpetrator had only been viewed by "Rebekah Vardy's account".
Mrs Vardy has continually denied leaking the stories in question to the press.
Mrs Rooney was christened "Wagatha Christie" by many people online as a result. Wag is a term used to describe the wives and girlfriends of footballers.
In court on Tuesday, Mr Rooney confirmed he had "never discussed it [the sting operation] really" with his wife and that "this trial is the first time I'm hearing almost everything in this case."
"It's been a long week," he said.
"For me and my wife, we don't want to be in this court. I have watched my wife over the past two years really struggle and become a different mother and different wife.
"It's been very traumatic for my wife."
He went on: "Whatever that judgment is, me, my wife and children can go on with our lives because it's not something we want to be part of."
The court also heard Mr Rooney had never had a personal relationship with any journalist from the Sun newspaper.
"Everyone knows the history between Liverpudlians and the Sun newspaper," he said, with reference to the coverage around the Hillsborough disaster. "I have never spoken to a Sun journalist on a personal level."
Mrs Vardy left court early again on Tuesday, as she did the day before, alongside her husband. But the Rooneys hung around to hear evidence from technical experts called by both sides.
Mrs Rooney's barrister David Sherborne last week told the court there has been a "widespread and significant destruction or loss of evidence" in the case.
Technical expert Ian Henderson, instructed by Mrs Vardy, told the court late on Tuesday he found it "surprising" there was an "absence" of WhatsApp messages between her and her agent Caroline Watt after the former tried to export messages from her device to her solicitors.
Matthew Blackband, an expert called by Mrs Rooney's team, stated his belief that files had been lost via a "manual deletion".
The trial is expected to conclude on Thursday.