I think the 38% number is definitely wrong, maybe even wrong by one order of magnitude.

I tried fact-checking and came up with this:

https://providencemag.com/2018/04/what-a-country-immigrants-serve-us-military-well

Few Americans realize that 65,000 immigrants serve in the US military today. That number includes some 18,700 troops who hold green cards (in other words, legal permanent residents who are not yet naturalized citizens). According to the Pentagon, about 5,000 such residents enlist each year.

Since the total number of the US armed forces is around 1.3 million, it follows that 38% is definitely wrong and the correct number of immigrants is likely around 5%.

I think the 38% number originates from service member naturalization stats or service member family stats and has been mis-interpreted. Can’t tell for sure, didn’t hire a spy to find out. :)

P.S. Edit: found a secondary source with similar data:

https://www.usafis.org/can-i-join-the-us-military-as-a-green-card-holder/

Approximately 8,000 Green Card holders join the US military each year and around 35,000 are currently serving on active duty.

Since green card holders aren’t the only variety of immigrant, the number 65 000 seems plausible.