If a married couple are both on the deed to their home and both are living in the home, can one kick the other out of the home?

Can a spouse kick their spous out of their home in which they are both on the deed?

In one word, no.

One of the parties CAN start a Court action (divorce/separation) and ask that the other party be removed.

In NY if violence is involved - and probably in other States, too - one party can get a restraining order keeping the second party out of the marital home but that is not an eviction or permanent order and the restraining order has to be followed by a Court action to remove the person.