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.

Following a delayed start 34 overs were bowled, in which Gloucestershire moved on from 31-0 overnight to 120-1.
After a long delay, the two teams came back out under the lights at 21:00 BST, but the visitors remained untroubled.
After 21 from Chris Dent, Bancroft got to 69 with Will Tavare still there on 43 as the visitors closed on 150-1.
Having begun the day with fears of following on, on his 50th first-class appearance, Tavare's unbroken stand of 88 with Australian Bancroft, who passed his previous season's best of 53, leaves Gloucestershire just 59 runs short of that initial target.
But against a Sussex side who are unbeaten in seven matches with Gloucestershire, there is little hope of the visitors improving on their record of just two wins from 18 visits to Hove.

Cheick Cisse, who won taekwondo gold, was given $85,000 (??64,000) and Ruth Gbagbi, who won bronze in the women's event, got $51,000.
Cisse's gold medal was the first in the country's history.
Last year, President Alassane Ouattara made similar awards to members of the national football team after they won the African Cup of Nations.
He did not reveal any details about the houses given to Cisse and Gbagbi, or their location.
More on this and other African stories
Mr Ouattara and his wife Dominique wore the medals as they posed with the athletes for photos at the presidential palace in the country's commercial capital, Abidjan.
Ivory Coast had only won one Olympic medal until this year.
Gabriel  Tiacoh won a silver medal in the men's 400m in the 1984 Olympics in Los Angeles.
He died in 1992.
Ivory Coast used to be one of Africa's richest countries but its economy has been devastated by years of conflict.
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.
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Cameron Bancroft made only his second County Championship half-century of the season as Gloucestershire reached 150-1 against Sussex at rain-hit Hove.
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Ivory Coast's two medallists from the Rio Olympic games have been given cash awards and houses by the president.