This isn't very clear, at least based on the published information that I could find. The AIM 5-1-9 says:
Use of FAA Form 7233-4 is recommended for domestic IFR flights and is
mandatory for all IFR flights that will depart U.S. domestic airspace.
7233-4 is basically an ICAO flight plan form, and the AIM says this about the alternates:
(b) Alternate and 2nd Alternate Aerodrome (Optional)
But it isn't clear - to me - if the word "optional" applies here to the entire line, or only to the second alternate.
Interestingly, the AIM also says this:
NOTE- Although alternate airport information filed in an FPL will be
accepted by air traffic computer systems, it will not be presented to
controllers. If diversion to an alternate airport becomes necessary,
pilots are expected to notify ATC and request an amended clearance.
That could mean that both the alternate and second alternate are optional, since there would be no point in requiring information that isn't even visible to ATC. However that's a guess and could easily be wrong (the data could be useful in other ways); I couldn't find any guidance on this on ICAO's site either. And airlines probably have their own internal rules on identifying and filing alternates as well.
A flight returning to the US would almost certainly use an ICAO flight plan, so whatever the correct interpretation of "optional" is here, it would apply to those flights too.
Anyway, regardless of what the regulations say, filing alternates does have the significant benefit of 'forcing' you to plan and review your route more carefully.