The Greek word "xenoi" is also translated as the English "host". Clearly Gaius is a believer.
Rom 16:21 ¶ Timotheus my workfellow, and Lucius, and Jason, and Sosipater, my kinsmen, salute you.
Rom 16:22 - I Tertius, who wrote this epistle, salute you in the Lord.
Rom 16:23 - Gaius mine host...