Answer: No. An individual who is convicted based on false testimony cannot sue the lying witness for civil (or money) damages.
In the American legal system, a witness testifying under oath, even falsely, is immune from civil liability for anything the witness says during that testimony. The reason for this is that our system of law operates best when witnesses are encouraged to testify freely and without fear of having to pay damages to someone because of what they said under oath. If a witness faced possible financial penalties for what he testified to under oath, he may be less willing to come forward and testify at all, even if he intended to tell only the truth.
Protecting witnesses who testify from later being sued also promotes another goal of our system: providing closure with a final decision in a case, once the appellate process is over. Allowing an endless string of lawsuits back and forth by parties to the same dispute would postpone the finality of the underlying case, clogging the courts and making it impossible for the parties to move on.
But, that does not mean that a person who intentionally lies under oath, and by doing so causes an innocent person to be convicted, can do so without fear of any negative repercussions. A witness who intentionally lies under oath has committed perjury and could be convicted of that crime. The crime of perjury carries the possibility of a prison sentence and a fine (paid to the government, not the individual wronged by the false testimony).
A person who falsely accuses you of a crime when he was not under oath could be sued for slander. Say a co-worker who is angry with you tells your boss and co-workers that you embezzled company money when you did not do so, and you got fired as a result. You could sue that person for slander.
Also, if the witness was not under oath and told a lie to the authorities that led to your arrest, the witness may have committed other offenses, like making a false police report.
And, a person wrongly convicted because of false testimony may be able to challenge the conviction on appeal based on the false testimony.
If you believe that you have been wrongly convicted because a witness lied under oath at your trial, you should contact a criminal defense attorney to find out if you can challenge your conviction. Be sure to engage someone who specializes in handling criminal appeals.