In the Untied States, if someone was caught claiming to have a college degree that they do not have on their resume, could they be criminally charged?
Yes! Some states have laws against it. RCW 9A.60.070, for example, makes it a gross misdemeanor in Washington.
(2) A person is guilty of knowingly using a false academic credential if the person ... falsely claims to have a credential issued by an institution of higher education that is accredited by an accrediting association recognized as such by rule of the student achievement council: (a) In a written or oral advertisement or other promotion of a business; or (b) With the intent to: (i) Obtain employment;
(5) Knowingly using a false academic credential is a gross misdemeanor.
If a state does not have a law specific to the issue, it depends on whether the statement is made under penalty of perjury under state law. It is not uncommon to have an employment application that asks for educational information and which you sign under penalty of perjury.
In addition, if the employer is a federal agency, for example, or if there is otherwise an argument that the resume is within jurisdiction of a branch of the Federal Government, the crime of making false statements could theoretically be used against a person.
In reality, in a state without a law specific to this issue, it is much more likely to be a really bad idea that will tank a person's career than it is to result in a criminal charge. Except perhaps if they do it during an FBI investigation or as part of a security clearance process.