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."