The tweet, which he later deleted but which has been circulated on social media, showed an image of a couple holding hands with a chimpanzee dressed in clothes with the caption: "Royal Baby leaves hospital".
The BBC 5 Live presenter was accused of mocking the duchess's racial heritage.
Baker claimed it was a "stupid gag".
The 61-year-old presented a Saturday morning show on the network.
The corporation said Baker's tweet "goes against the values we as a station aim to embody".
It added: "Danny's a brilliant broadcaster but will no longer be presenting a weekly show with us."
https://www.bbc.com/news/entertainment-arts-48212693