From what I can find, CARTO does not have a free map tile server. In other words, while it technically works, I think it's against their terms of service. They probably don't care if you're small, so I wouldn't worry about legal threats, but at some point the tile server might check the referer and break your app.
It looks like you can just use OpenStreetMap instead, although the resources are donated so there is no guaranteed Service Level Agreement: https://operations.osmfoundation.org/policies/tiles/