When using a magic item that can cast wish in other ways with a non-caster, like ring of three wishes used by fighter: if he/she failed his roll on the first wish would he be unable to use the other two?
It depends on the specifics.
Specifically for the Ring of Three Wishes; it says (DMG p. 193)
While wearing this ring, you can use an action to expend 1 of its 3 charges to cast the wish spell from it.
It is specific that you cast the spell; any consequences of doing so would apply to you. So, for example if you got the "can never cast Wish again" on the first try then the item is now useless to you except as bling.
This is a specific beats general situation; in general a non-caster can't cast spells; this item allows them to cast wish, except as modified (eg no material components that flows from being a magic item) normal rules apply.
I haven't looked but it may be the same for all magic items.