Is "can't" a euphemism or is it ambiguous?

Solution 1:

Neither. As I see it, it's just implied context that's omitted.

For instance, "I can't continue in this relationship" means something like "I can't continue in this relationship without significant loss to myself" or "I can't comfortably continue in this relationship", and this extra context is taken for granted. Similarly, "You can't embarrass me in public" = "You can't embarrass me in public [without inviting my disapproval]". Consider, say, "I can't jump off the tenth floor" where one does not usually bother to say "I can't jump off the tenth floor without dying".

It's not "I won't", but really a qualified "I can't" -- it's infeasible, or impossible under reasonable constraints.

Solution 2:

The primary uses of can are to express possibility and ability, with permission a third. Will expresses chiefly prediction and secondly volition. Should expresses mainly obligation, secondarily deduction and thirdly unreal and tentative meanings.

In your first example, the speaker uses can’t to express inability to continue in the relationship. We don’t know the reasons behind this inability, but they are clearly of an overriding, and presumably emotional, nature. Had the speaker said I won’t continue in this relationship we would have gained the impression that the emotional wherewithal was present but that other reasons had prevented the continuation of the hitherto happy state of affairs. In practice, I won’t continue in this relationship is likely to be followed by a clause beginning with, for example, unless . . .

If a speaker says You can't embarrass me in public, the implication is that the person addressed is unable to embarrass the speaker in public without certain consequences ensuing.