I'm playing Slay the Spire in Japanese, and often see like ４ダメージ and stuff like that, but I don't know how you'd actually say that aloud. There's lots of information about counting in Japanese on the internet but I can't find anything about how it applies to this specific problem. Would you say よんダメージ, よっつダメージ, しダメージ, or maybe something else?
It can be only pronounced よんダメージ.
ダメージ triggers no sound change. You effectively just put it after plain numerals (いち, に, さん, よん, ご, ろく, なな, はち, きゅう, じゅう...).
よっつダメージ is impossible unless you try to say like "give it four valid hits". When ダメージ means damage value, we only use itself as a pseudo-counter or without counters, such as よんのダメージ, ダメージよん.