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I was trying for a minimalist answer -- but I oversimplified.

For large populations, the sample size required for a given margin of error and confidence level doesn't change. I.e. -- if everything else is the same, the sample for a population of 350,000 would be the same as for a population of 350,000,000.

However, there is an adjustment for smaller populations -- and you originally asked about the 138,000 troops in Iraq. If 944 was the sample for the U.S. (assuming a simple random sample -- no stratification, or weightings) -- then you would need about 937 troops in your sample. Pretty much the same sample size, for practical purposes.

I just read the article carefully -- something I should have done first, but I was more interested in the comments.

My second post to you is correct for simple random samples. However, it does appear that, in this case, they did some stratification and weighting in order to be able to report differences between branches of the services, and regular vs. reserve troops. They might also have oversampled officers to see if there is a difference between the attitudes of officers vs. enlisted troops. This would obviously require a larger sample size.

These are only my guesses -- the article did not provide enough information to actually determine the sampling strategy. The margin of error reported would be for the overall statistics -- when they're reporting differences between branches, etc. there would be a considerably higher margin of error involved -- even though that is not stated.