The Staff of Healing has a lot of spells I wouldn't expect to see a wizard cast - Regenerate, Heal, etc, but I don't see anything in the rules on staves that precludes a wizard from using it.
Is any staff usable by any caster?
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The rules for Casting a spell from a staff spell out the requirements to cast from a staff. The relevant part of these requirements:
You can Cast a Spell from a staff only if you have that spell on your spell list, are able to cast spells of the appropriate level, and expend a number of charges from the staff equal to the spell’s level.
Also, you need to be the one who prepared the staff to cast from it:
A staff gains charges when someone prepares it for the day. The person who prepared a staff can expend the charges to cast spells from it.
To prepare a staff, you also need at least one of the spells the staff can cast on your spell list.
You can prepare a staff only if you have at least one of the staff’s spells on your spell list.
The list of Arcane spells (Wizards are arcane tradition casters) does not contain "Heal" or "Stabilize" (The spells that a generic Staff of Healing can cast), so the Wizard could neither prepare the staff nor cast spells from it.