Mr Henson has regularly criticised US President Donald Trump on social media. On Sunday, the television presenter suggested in a tweet that Trump is not Barron Trump’s father.
“I hope Barron gets to spend today with whoever his dad is,” Mr Henson, tweeted on Father’s Day.
Stephanie Grisham, the first lady’s spokeswoman, defended the president’s young son from Mr Henson’s remarks.
“Sadly we continue to see inappropriate and insensitive comments about the President’s son [Barron Trump],” the former White House press secretary told The Hill in a statement Tuesday.
Mr Henson also received outraged replies from other Twitter users after his tweet went viral.
One user wrote: “Hey look.. the left is attacking Barron Trump again.
“They can’t hide their hate for long I guess. Disgusting.
“I have ALWAYS stopped anyone on my timeline or show from attacking Sasha or Malia.”
Another user pointed out: “Why is Twitter allowing users to attack Barron Trump AGAIN?
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Mr Henson wrote: “With respect, I think you dissecting it is the issue.
“The joke was aimed at Trump and the mere mention of Barron’s name doesn’t mean it’s at his expense.
“Although I respect your right to take issue with it,” the father of two concluded.
Melania Trump has spoken out in defence of her son before after Stanford Law professor Pamela Karlan mentioned him during House impeachment hearings in December.
Ms Karlan said: “The Constitution says there can be no titles of nobility, so while the president can name his son Barron, he can’t make him a baron”.
The first lady took to Twitter to publicly rebuke the comments Ms Karlan had made before the House Judiciary Committee.
“A minor child deserves privacy and should be kept out of politics,” Ms Trump wrote.
“Pamela Karlan, you should be ashamed of your very angry and obviously biased public pandering, and using a child to do it.”
Ms Karlan later apologised for her remarks.