I am trying to see if this link is a full list. If not where can I find one?
No, this is a very short list.
This Wikipedia article puts the number of Jewish-American soldiers at about 500,000.
I can confirm that neither my grandfather or his brother (KIA), US soldiers serving in France during 1944 and 1945, are on the list that you have linked to.
It would be very difficult to find a full list because in the US, Jews are considered white, and large numbers of racial Jews are not Jews by religion. So I doubt there is anywhere on the internet you could find such a list.
For reference, most eligible men were drafted in WW2. So any given male born 1916-27 likely served in the war. You can look up census records for Jewish sounding names, but this will not be a comprehensive list because it doesn't consider unfit for service or crypto-Jews.