No. Just no.
There is no prohibition against a person with tattoos using a mikvah, either male or female. I have never heard of anyone being turned away from a mikvah, other than single women (but that’s a different discussion – premarital sex, not tattoos).
Getting a tattoo is prohibited, but there is no prohibition on what you already have. In fact, you are probably prohibited from removing it!
You are neither the first nor the last tattoo a mikvah attendent will see. I can reassure you about this because I have tattoos and have used mikvaot throughout the US and in Israel without comment. Tattoos are incredibly common in the BT, convert, and flirted-with-leaving-orthodoxy crowds.
I find that men are generally the ones most concerned about tattoos at the mikvah. After all, women have to go, but men choose whether to go to a mikvah. Even more, men can attend the mikvah in groups, and the other people using the mikvah with you may not be as rational, sane, or as observant of the laws of speech. As they say, don’t judge Judaism by the Jews. But the chances of a bad experience are slim.
Men, as Yom Kippur approaches, embrace the power of mikvah and don’t let concerns about tattoos stop you. (But remember that any piercings will have to be removed.)
Except when your tattoo is a cross tattoo. I had a whole tearful crisis around this before going to the mikvah that I fell apart in front of my rabbi about. http://twojewishmoms.blogspot.com/2013/03/liberation.html