Praise be to Allah.
It is not permissible to sell gold for gold unless two conditions are met: the exchange should take place in the same meeting in which the agreement was reached, i.e., hand to hand; and they should be equal in weight, because of the report narrated by Abu Sa‘eed al-Khudri (may Allah be pleased with him) who said: “Gold for gold, silver for silver, like for like, hand to hand. Whoever gives more or asks for more has engaged in riba, and the taker and the giver are the same.” Narrated by Muslim, 2971.
So it is not permissible to sell gold for gold of different amounts or deferred payment; no differentiation is made between white gold or other kinds of gold, because what is meant by white gold is nothing more than gold to which some platinum has been added to make it white. In fact it is still gold, but it is mixed with something else that did not change it from being gold. This has been explained previously in the answer to question no. 68039
An-Nawawi (may Allah have mercy on him) said concerning the words of the Prophet (blessings and peace of Allah be upon him), “Do not sell gold for gold or silver for silver unless it is same for same”:
The scholars said: This includes all kinds of gold and silver, both high and low quality, sound and broken, jewellery and ingots, and so on, whether it is pure or mixed with something else. There is scholarly consensus on all of that.
End quote from Sharh Muslim.
The correct way to do this transaction is to sell the ring for cash, then buy whatever you want of gold and other things with the cash.
And Allah knows best.