In a series of angry texts, Sir Gavin angrily accused the former chief whip Wendy Morton of seeking to “punish” MPs like him who were out of favour with then-premier Liz Truss by excluding them from the Queen’s funeral.
Rishi Sunak is now under fire for bringing Sir Gavin back into the Government, even though the Prime Minister had been warned the former education secretary was under investigation for alleged bullying.
Speaking on Times Radio, Mr Ross said he would sack any frontbencher found sending texts like those sent by Mr Williamson.
He said: "Yeah, it's not acceptable. I think the Prime Minister's spokesperson says that's not acceptable. And there's an investigation going on at the moment."
Pushed on whether he’d sack them, the Moray MP answered: "Yes, they wouldn't be in my front bench with the language that's been used.
"But I understand what the Prime Minister's saying, there's an investigation underway.
"And I don't know all the details, but I've seen the messages … and I wouldn't expect it from any of my colleagues in Holyrood."
It comes as Cabinet minister Grant Shapps described the correspondence as “completely inappropriate”.
The Prime Minister has now sought to clarify that while he was aware of a “disagreement”, he did not know the details of the exchange at the time he brought Sir Gavin back into Government.
Asked on Monday if Mr Sunak has full confidence in the Cabinet Office minister, the Prime Minister’s official spokesman said: “Yes.”
Pressed on why the Prime Minister gave Sir Gavin a job, Mr Sunak’s spokesman added: “Obviously he thinks he has an important contribution to make to Government.”
On whether Mr Sunak has spoken with either Sir Gavin or Ms Morton since the matter came to light, the Prime Minister’s spokesman said he was “not aware” of any talks, but would not normally get into details of private conversations.
Labour leader Sir Keir Starmer said Sir Gavin was “clearly not suitable” for the job, and claimed Mr Sunak was “weak” to appoint him.
Sir Keir told broadcasters during a visit to Imperial College London: “It is so disappointing that yet again, we’re having a discussion about the Prime Minister’s judgment, this time in relation to Gavin Williamson.
“He’s clearly got people around the Cabinet table who are not fit to be there. That is because he was so weak and wanted to avoid an election within his own party.
“I think the only way out of this, because these debates are going to go on because of the weak position the Prime Minister is in, I think we should say to the public, they should have a choice – do you want to carry on with this chaos, or do you want the stability of a Labour Government? That’s why I think there is such a powerful case for a general election.”
Asked if he was calling on Sir Gavin to be sacked, Sir Keir said: “I think that the Prime Minister has got people who are clearly not fit for the job around the Cabinet table.”