The veteran broadcaster posted a black and white photo of a man and woman holding hands with a suited chimpanzee after Meghan gave birth last week.
The 61-year-old captioned the tweet: “Royal baby leaves hospital,” prompting accusations of racism.
He was subsequently sacked by BBC Radio 5 Live, while Scotland Yard said the tweet was being “reviewed and assessed by specialist officers”.
The Metropolitan Police have now said it does “not consider that a criminal threshold has been met” and no further action will be taken.
A spokesman for the force said: “An allegation was received by the Metropolitan Police Service on Thursday, 9 May in relation to a tweet published on 8 May.
“Having reviewed the comments made, we do not consider that a criminal threshold has been met and as such we will be taking no further action.”
Baker deleted the controversial tweet shortly after it was posted, admitting it was “ridiculous, stupid and idiotic”, but insisted he was not racist.
A day after being fired from the BBC, he posted a series of messages apologising for the “outrage” he had caused.
The controversy had made it “one of the worst days of my life”, he said, adding that he had made a “genuine, naive and catastrophic mistake”.
He wrote: “The picture in context as presented was obviously shamefully racist. It was never intended so – seriously, who on earth would ‘go there’?”
But he hit out at the BBC, saying that the phone call telling him he had lost his radio show was a “masterclass of pompous faux-gravity”, adding that the corporation “literally threw me under the bus”.
A BBC spokesman said the tweet was a “serious error of judgement and goes against the values we as a station aim to embody”.
He added: “Danny’s a brilliant broadcaster but will no longer be presenting a weekly show with us.”
On Sunday, Baker was given a standing ovation at his first live show since being sacked by the BBC
He told the audience at the Theatre Royal in Nottingham: “When life deals you lemons, you chuck them at the people calling you racist.”
Prince Harry and Meghan, whose mother is African American, revealed last Wednesday that they had named their new arrival Archie Harrison Mountbatten-Windsor.