He also accused the first lady of attacking Hillary Clinton in 2007 by invoking a line she had said about fitness to run the White House.
The Obama campaign had denied the line referred to Mrs Clinton.
Mrs Clinton, meanwhile, accused Mr Trump of threatening democracy if he did not accept the election result.
"We know the difference between leadership and dictatorship, and the peaceful transition of power is one of the things that sets us apart," she told a crowd in Cleveland, Ohio.
"Donald Trump refused to say that he'd respect the results of this election. By doing that, he's threatening our democracy."
At his own rally in North Carolina, Donald Trump told supporters, "I see how much [Michelle Obama] likes Hillary."
"But wasn't she the one that originally started the statement, 'If you can't take care of your home,' right? 'You can't take care of the White House or the country?' Where's that? I don't hear that. I don't hear that."
The New York businessman was referring to a remark Mrs Obama made in 2007 while campaigning for her husband, who was running against Mrs Clinton.
The first lady said: "If you can't run your own house, you certainly can't run the White House".
Some critics asked whether the comment had been aimed at Mrs Clinton's relationship with her husband, former President Bill Clinton.
But the Obama campaign maintained the line was not directed at the Clintons and instead referred to their own struggle with parenting during a campaign.
"So, we've adjusted our schedules to make sure that our girls are first, so while he's traveling around, I do day trips," Mrs Obama continued in her 2007 speech.
"That means I get up in the morning, I get the girls ready, I get them off, I go and do trips, I'm home before bedtime."
Mr Trump was back on the campaign trail a day after he and Mrs Clinton appeared together at the Alfred E Smith Memorial Foundation charity dinnerin New York City.
The event, which takes place during election years, offers presidential candidates a chance to crack jokes about one another.
But Alfred E Smith V told CNN on Friday Mr Trump had "crossed the line and took it a little too far".
"The room did get a little uncomfortable," he added.
What happens next?
- The two candidates will spend the remaining 18 days before the election criss-crossing the US in their bid to persuade undecided voters. Expect to see lots of appearances in battleground states such as Ohio, North Carolina, Florida and Pennsylvania
- Voters will go to the polls on Tuesday 8 November to decide who becomes the 45th president of the US
- The new president will be inaugurated on 20 January 2017