List of navy seals

No, there is no public list of all current\former Navy SEALs. Their names are not classified as previously stated, it's just not all public.

If you want to verify somebody's claim that they are a Navy SEAL you could pull their Military Record using The Freedom of Information Act (FOIA) to do this you need to contact the National Personnel Records Center.

You cannot request this via email, you must submit a signed and dated letter or visit the building in person. Alternitively, you could ask Don Shipley, he is a retired SEAL who claims to be able to verify if somebody is or ever was a Navy SEAL or if they're lieing.