Among many other points made in his lengthy submission, Mr Doherty argues that Mr Johnson's mayoral powers to make policy decisions on matters such as housing, policing and transport pose a conflict of interests with his duties as an MP.
A Privy Council spokesman said the office was currently taking advice and would respond to the petition in due course.
Before the election, petitioners said that were Mr Johnson to be elected they would go to the High Court to argue a similar case – that Mr Johnson's role as head of the Mayor’s Office for Policing and Crime (MOPAC) prevents him from legally being able to serve as an MP.
They argue that heading MOPAC is essentially the same as being a Police and Crime Commissioner (PCC), and regulations prevent PCCs from simultaneously being MPs.
However, the Electoral Commission has rejected this argument, saying the two roles are different in law.
Sabrina Moosun, who also ran against Mr Johnson in the election, but as a candidate for the Communities United Party, filed an application for a court summons against him ahead of May 7, accusing him of electoral fraud.
However, this application has now been refused by a district judge, Westminster Magistrates' Court confirmed.
In her submission, Ms Moosun had argued primarily that Mr Johnson could not legally stand for election as an MP because, as London Mayor, she claimed, he was already a civil servant and a member of the legislature.
The Greater London Authority (GLA) monitoring officer, Ed Williams, had said in a letter to Ms Moosun's solicitor that Mr Johnson falls into neither category, dismissing her case as having “no foundation in law”.
Mr Johnson told getwestlondon: “The law is very clear on this, but obviously if they want to pursue it, they must. But there's no precedent and there's no legal obstacle, so I think let them get on with it."
A spokesman for Mr Johnson added: "As the independent GLA monitoring officer made clear in his response to Ms Moosun's solicitor back in April, there is no validity in her complaint. There is precedent for a Mayor being an MP – [Ken] Livingstone did that.
"In addition, the Mayor of London is not a civil servant, the GLA is not a legislature, the London Assembly is not a legislature, and the mayor is not a member of the Assembly."