It's not exactly what you're asking for, but <https://fly.io> is pretty neat. It's easy to get going and will deploy any docker images you want with a bit of their config added.
As for AWS ECS mentioned in the other reply -- it's great if you ever get it working, but it is a nightmare to learn and use.
Unfortunately, there's a huge gap between "single container" and "multiple, co-dependent containers". It's a far cry between services hosting a single container and ones offering to host entire stacks.
Right, I wasn't suggesting you could deploy with Docker Compose on fly.io. Hence why I prefaced my suggestion with "It's not exactly what you're asking for..."
And I think you're exaggerating the "huge gap" there. Fly.io can run multiple docker containers for you. Their docs are great so it wouldn't take much effort to learn how to create an equivalent or better setup that covers everything Docker Compose does (and more).
As for AWS ECS mentioned in the other reply -- it's great if you ever get it working, but it is a nightmare to learn and use.