You can see the requirements on the site American Airlines has arranged. A negative test result is still required.
The CDC's site directly addresses this issue:
What if I have had a COVID-19 vaccine or have tested positive for antibodies? Do I still need a negative COVID-19 test or documentation of recovery from COVID-19?
Yes, at this time all air passengers traveling to the US, regardless of vaccination or antibody status, are required to provide a negative COVID-19 test result or documentation of recovery.
Given the variety of vaccines available around the world, uncertainty about their effectiveness in preventing transmission, the difficulties around various proposed "vaccine passport" schemes, uncertainty about vaccine efficacy and protection against variants, and so on, it wouldn't surprise me if this remained the case for some time.