When you sue someone for a false accusation, you sue under the slander or libel laws of the state. Those laws generally require a person knowingly making a false statement to be liable for slander or libel.
Under § 1001, a statement is a crime if it is false, regardless of whether it is made under oath. In contrast, an oath is the hallmark of the three perjury statutes in Title 18. The oldest, §1621, condemns presenting material false statements under oath in federal official proceedings.