Can I sue my ex-girlfriend?

You can take her to the Small Claims Court (obviously that depends on how much you spent to pay off her loan!) Can you provide anything in writing to prove the loan? Maybe an email or any written agreement that discusses the loan. This can be a good evidence. If you don’t have any written communication, then find out bank or credit card statements that show you’ve provided your ex-girlfriend with the money. Moreover, you must be prepared to answer the questions for your case.

First of all, check out the Statute of Limitations in your state to sue a person based on verbal contract.
I would recommend you to consult a lawyer and he/she can guide you the best.
As Phil has said, usually, you can file a claim at the Small Claims Court in such cases.
From next time, when you lend an amount, it’s better to make a written contract. I know, it’s not always easy to do it among friends and family members; but, you have no choice when the other person doesn’t keep the promise.