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.”
Baker’s Saturday Morning show on BBC Radio 5 Live won him a Sony Gold award for Speech Radio Personality of the Year in 2011, 2012 and 2014 and a Gold Award for entertainment show of the year in 2013.
His irrepressible style made him one of the most popular radio presenters of his generation and saw him described by one writer as the “ultimate geezer”.
Baker was also a successful magazine journalist, scriptwriter and TV documentary maker.