Is the CAA medical class 1 certificate equivalent to the FAA class 1 medical certificate? How so?
Or does one have to convert to the FAA license in order to achieve the medical certificate?
The CAA medical (UK or any other country with a CAA) is not equivalent to an FAA medical, you have to get an FAA medical to get a FAA license. The classes are similar but they cannot be used in place of the other. In the UK you can get an FAA medical from several AMAs.