"I gave a letter to my girlfriend."
No, あげる is used to give something as a present or a service. Letters are not presents. We say 手紙を書いてあげる (write a letter as a service), otherwise if it is handover the letter, it is 手紙を渡（わた）す. If it is by post, 手紙を送（おく）る
I feel more natural if we use the honorific though - 手紙を差し上げる