To to mention its pretty set in stone that Gio knows how felix feels.
Be it from the fact felix striaght up confesses in then1st season to how he constantly gets angry for Gio.
There is always some hidden story that we don't learn about until something has already happened or if it comes up via another character.
Be it a character is just incredibly insecure or something is being kept secret from the MC.
While gio did just say "I dont want him leave me" it could mean everything from he is insecure about the relationship. Or...sense they had another spat about a medical procedure right before he mentioned it. It could be something he isn't telling anyone.
Giovanni refuses the treatment
For 2 possible reasons.
1) It hurts felix or himself some how. And he will not risk that. And he is keeping quiet about it on purpose.
2) He still thinks his condition is the only thing keeping Felix by his side. Healing his problems makes him.believe Felix will no longer want to stay with him.
Its most likely the 2nd one to be honest lol