Dr Hudson, 29, said she could not reconcile her union's new "extreme" position on abortion with her lifesaving work.
She said: "As a doctor who looks after babies from 23 weeks, I am really happy that the babies we look after have a complete right to life enshrined in law.
"We wouldn't dream of withdrawing care from a baby in one of the incubators at our neonatal unit just because parents didn't want it.
"Just because a baby is in a womb, rather than an incubator, I still believe that it deserves legal protection over its rights."
Dr Hudson added: "The BMA voted that abortion should be regulated as if it was just another medical procedure, but I really believe that abortion is massively different from an appendectomy."
Dr Matthew Knight, a hospital consultant in Watford, is also quitting the union.
He said: "While I do believe the BMA on the most part works tirelessly for both the interests of patients and of its members, I have been concerned for some time that its agenda has been increasingly hijacked by people with more extreme views."
Dr Richard Loveless, a GP from Somerset, said he was "very disappointed" by the vote and was also quitting the BMA after 35 years.
Abortion is technically illegal in Britain unless it meets criteria under the 1967 Abortion Act.
There is an upper time limit at 24 weeks, except in cases of severe foetal abnormality, and two doctors must give consent for the procedure.
A BMA spokesman said: "It is important to note that doctors also voted for abortion to be regulated in the same way as other medical treatments, therefore decriminalisation would not mean deregulation.
"The debate and the BMA's new policy only relate to whether abortion should or should not be a criminal offence. The BMA has established policy on these issues - such as supporting the 24-week limit - which will remain unchanged."