Concede or Escalate
The longer you and your teammate stew in stalemate, the more damaging it is to your relationship. If alternatives haven’t gotten you unstuck, your options are to either concede or escalate.
Weigh the cost of just approving the changes. You can’t build quality software if you casually accept low-quality code, but you also can’t achieve high quality when you and your teammate fight so bitterly that you can no longer work together. How bad would it really be if you approved the changelist? Is it code that could potentially destroy critical data? Or is it a background process where, at worst, the job will fail and require a developer to debug it? If it’s closer to the latter, consider simply conceding so that you can continue working with your teammate on good terms.
If concession is not an option, talk to the author about escalating the discussion to your team’s manager or tech lead. Offer to reassign to a different reviewer. If the escalation goes against you, accept the decision and move on. Continuing to fight it will drag out a bad situation and make you look unprofessional.