No they don't anymore. It is automatic and has been since this year. And no...the sitting President can not just call them up. It has to be passed by Congress. You can thank Milton Friedman for that. Congress would never agree to that because they only care about getting paid by tax payers. And absolutely everyone knows that conscription never works...ever.
We had a continuous draft(conscription) from 1947/8 to 1973, the last draft being 1972. Men were drafted during peacetime as well. It was in 1980 that kids needed to go a sign into selective service.
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