My ex wife is making all sort of embarassing and insulting comments about my character which aren't true online. Can I sue her?

You can definitely sue her: Internet defamation is defamation; there is no special protection for making defamatory comments online. Defamation is the public making (and the Internet is public) of a false factual statement that puts someone in a negative light or damages their reputation (and being accused of child pornography is almost as negative as it gets). Opinions are not actionable—your ex-wife could freely say, “I think that my ex-husband was the worst husband on Earth and a woman would have to be crazy to stay with him.” However, once you state something that purports to be fact, such as that someone reads, collects, etc. child pornography, that could result in liability.

The truth is a defense to a charge of defamation, so if is in fact that case that you read or collect child pornography, you could not recover from you ex-wife for defamation. (Though there may be other claims you could bring for impersonating you online.) It would be her responsibility to raise that as a defense and prove it, if you sue her and it goes to trial.

However, you cannot sue Facebook. There is a law called the Communications Decency Act which protects an “interactive computer service” provider—like Facebook—from liability for statements or content posted on it by other people. Section 230(c)(1) of the Communications Decency Act provides in part: “No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider.” The Act is designed to protect providers from the statements of their contributors or posters. However, you should be able to at least force Facebook to take this content down.

You should consult with an attorney who handles defamation cases; from what you write, it would be worthwhile exploring further the strength of your case, what it might be worth, and what steps you should take both to protect your reputation and to recover any compensation to which you may be entitled.