Mr Braid's case could test the constitutionality of the new law in the southern US state, which took effect 1 September.
The first suit against Mr Braid was filed with a district court in Texas on Monday by a self-described "disbarred and disgraced" former lawyer in the neighboring state of Arkansas.
"I acted because I had a duty of care to this patient, as I do for all patients, and because she has a fundamental right to receive this care," Mr Braid, who has been practicing medicine for 50 years, wrote in the Post.
"I fully understood that there could be legal consequences - but I wanted to make sure that Texas didn't get away with its bid to prevent this blatantly unconstitutional law from being tested," he added.
In the landmark 1973 case Roe v. Wade, the US Supreme Court guaranteed the right to an abortion so long as the fetus is not viable outside the womb, which is usually not until the 22nd to 24th week of pregnancy.
But the court, which was shifted to the right with the confirmation of three conservative justices nominated by Donald Trump, refused by a 5-4 margin to block the Texas law from going into force.
The bill passed by Republican lawmakers in Texas, the country's second-largest state, allows members of the public to sue doctors who perform abortions after six weeks or anyone who facilitates the procedure.
First civil suit
The complaint against Mr Braid was submitted by Oscar Stilley, who is currently serving a 15-year sentence for tax fraud under home confinement.
Abortion providers and others seeking to protect a woman's right to an abortion generally file suit against state prosecutors who are seeking to enforce restrictive abortion laws.
But Texas managed to avoid close judicial scrutiny by the Supreme Court because of the way its law is framed, making it everyday people rather than prosecutors who initiate legal proceedings over abortions.
Mr Braid's admission means that if he is prosecuted he could contest the constitutionality of the Texas law and force a court to rule on whether it violates Roe v. Wade.
The Justice Department has also filed suit against Texas, following through on a pledge by Democratic President Joe Biden to fight attempts by Republican-led states to restrict abortion.
And the Supreme Court announced on Monday that it would hear a challenge on 1 December to a Mississippi law that bans nearly all abortions after the 15th week of pregnancy.
The court ruled on the Texas law without hearing oral arguments, and the Mississippi case will be the first abortion case argued before the court since Trump named three justices to the panel, giving conservatives a 6-3 majority.