Now we have a public accommodation restricting your ability to engage in business via a third party.
You would have to sue the place that is refusing to admit you to practice on grounds that to do so violates the Unruh Act (Civil Code 51), the Contracts in Restraint of Trade Law (Bus. & Prof. Code 16600), and maybe the Sherman Antitrust Act (15 U.S.C. 1).
The goal would be to show that the various places of business that require your board certification are engaged in restraint of trade and are discriminating against you on the basis of age -- because the organization that certifies your ability to obtain admission to a particular facility arbitrarily creates rules that prevent you from reasonably practicing your profession -- a profession that is supposed to be governed solely by the Medical Board of California.
That's my first pass at the possible cause of action. This could be a very costly legal action -- but, if there are other physicians in similar circumstances, you might be able to make it into a class action. And, a class action filled with physician-plaintiff's could really make a large law firm's collective palms start to sweat with greed.
You'll have to shop the case. Maybe you, or some of your colleagues already have connections with a high-powered law firm. If not, then for a possible referral, see this link
Please let me know if I can be of further assistance.