Richard Cushworth and his Salvadoran wife Mercedes Casanella say a DNA test has confirmed the boy they are raising is not their son.
Local media say their child has now been found, after testing other babies born at the hospital on the same day.
The doctor, who denies any wrongdoing, is being prosecuted by the authorities.
At a hearing in the Central American country on Monday, a court ruled the case against Dr Alejandro Guidos should proceed and ruled that he could not leave the country. It also ordered that a new DNA test should be carried out following a request from Dr Guidos' lawyers.
Mr Cushworth and Ms Casanella, who attended the hearing, had originally travelled to El Salvador in May to have their baby in his mother's homeland.
But Ms Casanella says she became suspicious because the skin colour of the baby she was handed to take home was darker than that of the boy she saw after giving birth and their features were different.
When the couple returned to their home in the US they eventually decided to have a DNA test which they say confirmed the baby was not theirs.
Mr Cushworth, who is originally from West Yorkshire, said: "God has given us this child and somehow, somebody has taken him from us, and we want him back.
"It's a horrible situation for me, for her (his wife) for my family, her family. A child is an experience you have for a lifetime... this is a life-long injury that's very, very deep, and it's horrible."
Francisco Meneses, the couple's lawyer, said: "We don't have anything against the people who were involved during the baby's birth, but we want all these people to put their hands on their hearts because from the doctor who performed the surgery, the paediatrician, anaesthesiologist, and the two nurses who were in the delivery room, it's very important for them to tell us what happened."
Ms Casanella says she is prepared to continue looking after baby Jacob if his mother cannot be found. She says she is not seeking financial compensation but that she will not rest until she gets her baby back.
A British father and his wife have accused a doctor at a hospital in El Salvador of swapping their newborn son at the request of child traffickers.