US Airlines adopt a relaxed approach when it comes to flying pregnant women. Most of the airlines do allow pregnant women to fly without requiring a doctor certificate for most part of their pregnancy period. Some of the very minimum restrictions which they have imposed are given below. 

Spirit Airlines (NK)

Spirit Airlines requires women in 8th month of pregnancy to provide a doctor certificate during check-in.

Frontier Airlines (f9)

Frontier Airlines also requires that women in the 36th week of pregnancy shall have a doctor certificate. 

United Airlines (UA)

United Airlines does not need the medical certificate before the 36th week of pregnancy. However, this is required during and after the 36th week. Further, it shall not be older than 72 hrs before the flight departure time.

Delta Airlines (DL)

Delta Airlines adopts the most liberal approach as it does not require any certificate. However, it does advice to go for a check up with the doctor if flying after 8th month. 

Alaska Airlines (AS)

Alaska Airlines does not impose and limitation or restriction on pregnant women. However, it does require that the passenger shall consult doctor before flying.

Hawaiian Airlines (HA)

Hawaiian Airlines does not allow expectant mothers to fly 7 days before and after the scheduled departure date of flight. If they have to fly in this time period, then a certificate from a doctor, not more than 3 days old, is required to be furnished.

American Airlines (AA)

American Airlines adopts a different approach for its domestic and international flights, when it comes to carrying expectant mothers. 

When flying on domestic flights, they are not allowed to fly within 7 days of departure time of flight (both pre- and post- departure time).

In case of the international flights, if the due date of delivery is within 4 weeks of scheduled departure time of flight, medical certificate is required.The certificate shall not be more than 48 hrs old.

jetBlue (B6)

When flying jetBlue airlines, if delivery date is within 7 days, then doctor certificate in mandatory. It shall not be more than 72 hrs before the departure time of flight. 

Southwest Airlines (WN)

Pregnant women are not allowed to fly beginning from the 38th week of pregnancy. Doctoral consultation is recommended.