Whether it’s a physical affair or an emotional affair, does not matter.
If in the aftermath there has been no consequence great enough to make them apologize, they are getting away with it.
There will be no meaningful progress toward healing until they apologize.
The initial screaming wailing sobbing fit when you catch them red-handed doesn’t count. That’s just so you don’t kill them on the spot. They need to be not currently in a state of panic when they apologize.
You can try moving forward without their apology, but that’s more to do with your hope that everything will get better than your actual chances it will.
Sometimes there is nothing that will make them apologize.