Just because they have a tattoo?Probably,but with caveats.
If the tattoo is racist,hateful or otherwise offensive and/or inappropriate,maybe not,depending on their attitude towards it.They made an idiot mistake that they now profoundly regret and intend to deal with as soon as they can afford the cover-up or laser removal?Probably worth giving them the benefit of the doubt.
Otherwise,if you break up with someone just because they have chosen to decorate their skin and for no absolutely other reason,that probably says more about you than the tattooed person.And not in a good way.