Chris Heaton-Harris said on Thursday that he had extended an invite to dinner to the Sinn Fein party president, stating: "What happened yesterday happened yesterday. I'm much more focused on the big ticket items."
Responding to the dinner invitation, Ms O'Neill said: "We're not interested in dinner."
Mr Heaton-Harris has also said it may not have been wise to have Ms McDonald who is the leader of the opposition in the Republic, at discussions around the NI Protocol because she is a "representative of a parliament in an EU member state".
The UK Government and EU are currently involved in negotiations to resolve differences over the post-Brexit trading agreements in Northern Ireland.
The row led to Sinn Fein and the SDLP refusing to attend the meeting on Wednesday and has overshadowed a series of talks to discuss how to resolve issues caused by it.
The Government has also suggested that due to diplomatic protocol Mr Cleverly could not meet Ms McDonald before he had met his counterpart in Ireland, Micheal Martin.
But Mr Martin said he would have had no issue with the Sinn Fein leader attending the meeting, and said: "We certainly got no heads-up in relation to that at all, and I think it needs to be stated.
"I would have had no difficulty that Mary Lou McDonald was at that meeting but that's a matter for the Foreign Secretary."
Speaking on Thursday, Ms McDonald said: "All of us know that the way that we make progress is together, that's how this works, that's what the history of the last 25 years reflects, so any idea of excluding anybody, excluding the leader of any party, needs to be scotched and knocked on the head now."
She also confirmed she had written to Prime Minister Rishi Sunak about the issue.
Responding to the controversy after noting his dinner invitation to Ms McDonald, Mr Heaton-Harris said: "Sinn Fein were invited, it is a shame (Sinn Fein vice president) Michelle (O'Neill) didn't come along because it was an update on the protocol discussions.
"There are many factors which go into the thought process. One, to be quite frank, is that the UK Government is negotiating with the European Union.
"We wanted to update Northern Ireland parties on that negotiation and, with the greatest of respect, Mary Lou is a representative of a parliament in an EU member state.
"That might not have been seen as a wise thing to do."
Alliance Party deputy leader Stephen Farry was asked about Mr Heaton-Harris's contention that it would have been unwise to invite Mrs McDonald to Wednesday's meeting.
"I think he needs to back down from that," he said.
"The Government made a mistake in terms of how they handled the talks yesterday, their position on that is not tenable.
"They need to recognise the subtleties of the situation in Northern Ireland and the reality that Sinn Fein is organised on an all-Ireland basis.
"The Government need to de-escalate their language and they need to climb down and find a means of ensuring that we have proper inclusive talks at the next opportunity."
Ulster Unionist leader Doug Beattie said he hoped that all political parties from Northern Ireland would take part in future talks with the Government: "I wanted Sinn Fein to be there and I wanted the SDLP to be there.
"I wanted to hear what they had to say. I wanted us all to work together.
"And I've said this before, boycotts do not work of any shape or form. So whether it's the DUP boycotting the executive or any other party boycotting talks, it simply won't work."
But DUP leader Sir Jeffrey Donaldson made the point that Ms O'Neill, Sinn Fein's Stormont leader, was invited to the talks and decided not to go.
He told the BBC: "If we are into a situation where Michelle O'Neill can only attend meetings with UK Government ministers if she is accompanied by the leader of Sinn Fein in Dublin, if she has to have a minder with her, that has very serious implications for future arrangements in Northern Ireland if Michelle O'Neill were to become the first minister."