Is it permissible to exchange a white gold and diamond ring directly for gold from a merchant, and what is the correct way to do so if it is not permissible?

It is not permissible to sell gold for gold except under two conditions: mutual possession (hand-to-hand exchange) at the session of the contract, and equality in weight, regardless of its color or quality. The correct way to deal with it is to sell the gold for money, then buy what is desired with that money.