Complaint, filed in state court in New York, is here. It includes claims for Tortious Interference with Prospective Business Relationships (because of the effects on his endorsement opportunities) and Tortious Interference with Existing Contracts (because of the effects on his contract with the Yankees), all based on MLB's investigation cum vendetta against him.
I doubt this goes anywhere. Tortious interference is incredibly difficult to prove, particularly in requiring highly unlawful or tortious misconduct; baseball's actions, while skirting some ethical lines, does not seem to reach that level. There also is a good chance the claims are precluded by the CBA and federal labor law, since they functionally seeks to correct and overturn punishment imposed on him by baseball pursuant to its institutional arrangements, even when disguised as claims for tort damages (see Jonathan Vilma's defamation suit against the NFL). The irony is that it was a claim for tortious interference that MLB brought against Biogenesis, in an effort to obtain documents in discovery, which started all of this. The Rodriguez complaint labels a sham and identifies that as one of the tortious acts.
Update: A-Rod is not done. He filed a separate action for medical malpractice against the Yankees' team physician New York Presbyterian/Columbia University Medical Center, arising from the failure to diagnose his torn labrum last October.