Am I obligated to pay the translation costs on behalf of my friend, the doctor, who refused to pay them to the translator, knowing that I only coordinated between them to help them both?

If you clarified to the translator that you are an intermediary between him and the doctor and did not guarantee his fee, then you are not obliged to pay it, because you are a trustworthy agent. However, if you did not inform him of that, or if you guaranteed his fee, then you are obliged to pay it. Reconciliation is advised to resolve the dispute between you.