If my newborn just had his first MMR shot and he’s running a fever, should I call the doctor?

It Depends. The question is confusing. Newborns don't get an mmr. It is sometimes given as early as 6mo to people traveling in risk areas, but not newborns. Any newborn with a fever should be evaluated. If your child is older, a fever before 4-7 days is likely unrelated to the vaccine and means an illness not evident when the shot was given. Call and review the findings with your doctor.
It Depends. If the fever is less than 105 degrees or 40 degrees centigrade, and the baby is acting fairly well, then comfort treatment is ok such as tylenol (acetaminophen). If it is a reaction to the immunization, then call if the baby is acting sick.The reaction to the MMR is usually 10-21 days after the shot, but can be as soon as 4days.
Yes it's not the MMR. The initial MMR vaccine is typically given from 12-15 months of age. A booster dose follows at 4-6 years old. Mmr is a live virus vaccine; therefore the fever, rash, or fussiness attributed to vaccine administration does not occur with MMR until 7-12 days after the injection--unlike other immunizations. So, if there is fever just after the MMR shot, then this is caused by something else. Call.
Probably. Remember, MMR can not be given until at least one year. If your infant develops a fever (over 101 degrees) after any vaccine, and you are concerned, contact your doctor for advice. For the MMR vaccine, some children will get a mild fever and/or rash, but generally not until 10 days after receiving the vaccine.