Asked about the tweets at the briefing, White House deputy press secretary Sarah Huckabee Sanders said Trump doesn't care what people call the executive order, does not know whether a third "much tougher" version of the travel ban is being considered, and asserted that extreme vetting already "is taking place".
This was the second executive order President Trump signed on the matter.
The original order from January and a revised version have been caught up in the courts.
He confirmed to the Washington Post on Monday that the Twitter account belonged to him.
Instead, though, the President sent six tweets on Monday morning about the mayor of London, his travel ban and his Justice Department. Trump is known for making statements that are often inflammatory, or in contrast with policy positions and previously stated administration goals, on his Twitter account.
"These tweets may make some [people] feel better, but they certainly won't help [the Office of the Solicitor General] get 5 votes in [the Supreme Court], which is what actually matters", Conway tweeted.
Told by Guthrie that Trump's false claim about Khan was his "first tweet" - it was actually his fifth, following a retweet of the Drudge Report, a push for his contested "travel ban", a pledge of support to London, and another missive about USA security and not being "politically correct" - Conway shot back, "I'm not going to allow, on a day-and-a-half after terrorists did it again - whether they're ISIS-inspired or ISIS-directed, they're savage murderers". In an interview, Jadwat said the president's tweets "seem to undermine the picture the government's been trying to paint".
Jadwat concluded with what amounted to, more or less, the mirror image of the concern Conway expressed about Trump's tweets. Trump's tweets and anti-Muslim rhetoric have repeatedly been cited by federal judges who ruled against the administration's executive order, which would halt immigration from several Muslim-majority countries.
He added: "We don't need the help but will take it!"
Earlier on Monday, Kellyanne Conway said the media had an "obsession" with covering Trump's tweets.