Yes, with some exceptions.
Once you have a signed marriage license you’re legally married. The only exception I can think of is if that license isn’t accepted in the state or country you live in, but you’re still legally married where the license was issued.
For instance, gay marriage wasn’t always legal in America, so two men who were legally married in the Netherlands, were not legally married in the US, but a straight couple was considered legally married in the US.
Some countries may not even acknowledge straight marriages reciprocally, but I think most do.
Edit: once you complete the marriage license (all necessary signatures, necessary signatures vary by state in America, but all states acknowledge the other states’ marriage certificates) it gets filed and then you will be issued a marriage certificate.