It should be, an unlimited license is an unlimited license. But the only people who can really answer your question are your program and your GME office. Every institution has different moonlighting rules so you should definitely check there first.
I am not an immigration lawyer. The J-1 (Exchange) visa is specifically for training at the sponsoring institution. The terms of the visa and allowed work is detailed in your institution's filings with INS. Your visa is restricted to the terms of those filings which generally include only work/training within the sponsoring institution.
An unrestricted medical license will allow you the possibility of obtaining moonlighting work in medicine, but your visa may not. Check with your immigration lawyer or your institution to determine what, if any, work you can perform for pay outside of the specific training program.
As you know, undocumented workers are a major controversy in the US right now. I'd bet your visa will not allow you to work outside of medicine and may not allow you to work at your neighborhood clinic for pay. INS has become extremely strict in enforcement issues. For visa violations, people have been deported and once an INS action occurs, forever denied reentry into the US. Check with your program and your lawyer to make sure it is legal before you do it. This is not something you want to mess up on.
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Here are the rules (22CFR 62.62.16) Employment
(a) An exchange visitor may receive compensation from the sponsor or the sponsor's appropriate designee for employment when such activities are part of the exchange visitor's program.
(b) An exchange visitor who engages in unauthorized employment shall be deemed to be in violation of his or her program status and is subject to termination as a participant in an exchange visitor program.
(c) The acceptance of employment by an accompanying spouse or minor child of an exchange visitor is governed by Immigration and Naturalization Service regulations.