Harry and Meghan, whose mother is African American, revealed on Wednesday that their new arrival was named Archie Harrison Mountbatten-Windsor.
Radio DJ Baker made a joke about the birth, tweeting a black and white photo showing a well-dressed man and woman holding hands with a suited chimpanzee.
He later posted on Twitter: "Just got fired from BBC 5 Live. For the record - it was red sauce. Always," referencing a game on his show.
Baker had captioned the image: "Royal baby leaves hospital".
Following a backlash and allegations of racism, he deleted the post and apologised.
He said: "Sorry my gag pic of the little fella in the posh outfit has whipped some up. Never occurred to me because, well, mind not diseased.
"Soon as those good enough to point out its possible connotations got in touch, down it came. And that's it."
The 61-year-old, who presents a weekly show on BBC Radio 5 Live, apologised again on Wednesday evening, saying the joke was intended to poke fun at the royal family.
He said: "Once again. Sincere apologies for the stupid unthinking gag pic earlier. Was supposed to be joke about royals vs circus animals in posh clothes but interpreted as about monkeys & race, so rightly deleted.
"Royal watching not my forte. Also, guessing it was my turn in the barrel."
A BBC spokesman said: "This was a serious error of judgment and goes against the values we as a station aim to embody. Danny's a brilliant broadcaster but will no longer be presenting a weekly show with us."