Can I pick up and keep my child until the parent with physical custody is out of the hospital?

No, not without the permission of the hospitalized parent, or a right of first refusal to provide care in your court orders. Otherwise, it is the right of the custodial parent to determine who cares for the child when the custodial parent is unable to do so. Or you can file an emergency motion to have the child placed with you until the other parent proves fitness (which likely depends on why the other parent has been hospitalized).