This is an ultra-low, if any risk. There's absolutely no reason to be concerned, particularly if you went to a reputable tattoo shop. But really, there's no need to be concerned at all. Greater risks exist for bacterial infection (which you obviously don't have) and HepC/HepB from an incident like this.
Really, I wouldn't worry about it at all. If you're in a developed country, chances are you were given a 4th generation test, which is conclusive at 28 days. NO ONE recommends testing beyond 3 months, using even previous-generation tests, so if you can't sleep over this, test one more time 90+ days after the event and put it totally behind you. I cannot emphasize enough how low a risk this would be, though.
Your situation involves personal contact with an object in air (tattoo needle ). You will be happy to learn that you had no risk, because you can't get hiv from personal contact except unprotected penetrating vaginal or anal, neither of which you did and you didn't share hollow needles to inject with which is the only other way to acquire hiv - there are only 3 ways to get hiv. Analysis of large numbers of infected people over the 40 years of hiv history has proven that people don't get hiv in the way you are worried is a risk.
HIV is a fragile virus in air or saliva and is effectively instantly dead in either air or saliva so the WORST that could happen is dead virus rubbed you, and obviously anything which is dead cannot live again so you are good. Blood and cuts would not be relevant in your situation since the hiv has become effectively dead, so you don't have to worry about them to be sure that you are safe.
There is no reason for a person to test when they are safe. The advice took into consideration that the other person might be positive, so move on and enjoy life instead of thinking about this non-event. hiv prevention is straightforward since there are only 3 ways you can become infected, so next time you wonder if you had a risk, ask yourself this QUESTION. "Did I do any of the 3?" Then after you say "No, I didn't" you will know that it's time to move on back to your happy life.
No one got hiv from what you did during 40 years of hiv history and no one will get it in the next 40 years of your life either. You can do what you did any time and be safe.
If you informed him this was about a tattoo needle, consider finding a better doctor that doesn't waste your time.