Assuming that the documents were either true, or Manning reasonably believed that they were true, there would be no cause of action for defamation.
Many of the documents disclosed would have been confidential in some sense, but usually a violation of a confidentiality statute has a criminal sanction associated with it, but does not carry with it a private cause of action – in part, because conceptually, the party actually harmed is considered to be the government and not the person about whom information is revealed.
It is also possible that Manning could utilize the state secrets privilege as a defense and have such a suit dismissed on the grounds that a full and adequate defense of the claims would require the disclosure of official state secrets. For example, if a covert agent were murdered due to a wrongful disclosure of information, usually official disclosure of the fact of being a covert agent would be required to prove the case, and that evidence would be barred by the state secrets privilege, effectively barring the lawsuit entirely.
Constitutional claims of privacy violations under the 4th Amendment generally relate to the wrongful acquisition of information and not its wrongful disclosure. The constitution bars unreasonable search and seizure, not unreasonable disclosure of information.
The only privacy tort that might be applicable is "Public disclosure of embarrassing private facts." (A sister privacy tort, Intrusion upon seclusion or solitude, or into privacy affairs, is directed at the collection of data phase and not the dissemination phase).
See Restatement of the Law (Second) of Torts, §§ 652B and 652D.
But, this tort raises serious First Amendment concerns and has not been widely adopted. Realistically, this tort is unconstitutional in the absence of an affirmative contractual or quasi-contractual duty not to reveal facts that runs to the person making the disclosure, and in general, Manning would not have that kind of relationship.
The classic public disclosure of embarrassing private facts case would involve a lawyer's or psychotherapist's revelations about a client.
Also, in the case of the public disclosure tort the basis for damages is largely personal emotional distress and violation of trust, as opposed to damage to reputation, per se. The requirement is that the disclosure be embarrassing or breach of contract, not that it harm someone's reputation since you have no legal right to a reputation that differs from the truth.