The Church does not oppose tattoos in principle.
Most people who say it does cite, “You shall not make any cuttings in your flesh on account of the dead or tattoo any marks upon you: I am the LORD” Lev 19:28.
Catholics remain bound to the Torah’s moral law. The Catholic Church places great emphasis on the Ten Commandments as both moral and natural law. Of the Catechism’s 2,865 numbered paragraphs, 505 of them (more than 1 of every 6) (§ 2052–2557) address the Decalogue.
However, the Torah mitzvah against tattoos belongs to the ceremonial law, which does not bind Catholics, as Rabbi Paul told us, “Now if perfection had been attainable through the Levitical priesthood (for under it the people received the law), what further need would there have been for another priest to arise after the order of Melchizedek, rather than one named after the order of Aaron? For when there is a change in the priesthood, there is necessarily a change in the law as well” Heb 7:11–12.
After all, any Catholic who argues that the law against tattoos binds us also has to argue that the rest of the ceremonial law binds us, such as, “You shall not let your cattle breed with a different kind; you shall not sow your field with two kinds of seed; nor shall there come upon you a garment of cloth made of two kinds of stuff” Lev 19:19.
The moral and ceremonial laws appear together in the Torah. This law against mixing two kinds is the very next verse after, “You shall love your neighbor as yourself” Lev 19:18, which is most definitely moral law Mt 22:39.
We Catholics find God’s law for us in Church teachings. The Catechism says not a single word about tattoos. Neither does the Compendium. Neither does the Compendium of the Social Doctrine of the Church.