The Â£230m development consists of 11 wind turbines.
Mr Trump claims they would spoil the sea views for golfers on his course at Menie.
He responded: "We will put our future plans in Aberdeen on hold, as will many others, until this ridiculous proposal is defeated."
Mr Trump said: "We will be bringing a lawsuit within the allocated period of time to stop what will definitely be the destruction of Aberdeen and Scotland itself.
"This was a purely political decision.
"We will spend whatever monies are necessary to see to it that these huge and unsightly industrial wind turbines are never constructed."
He had previously threatened not to build a hotel at the resort if it went ahead.
Ministers said they were committed to a "successful and sustainable" offshore wind sector.
The European Offshore Wind Deployment Centre (EOWDC) project is a joint venture by utility company Vattenfall, engineering firm Technip and Aberdeen Renewable Energy Group (Areg).
The project is aimed at providing enough energy for more than 49,000 homes.
The Scottish government said the centre would allow offshore wind developers and supply chain companies to test "cutting edge" wind technology before its commercial deployment.
Energy Minister Fergus Ewing said: "Offshore renewables represent a huge opportunity for Scotland; an opportunity to build up new industries and to deliver on our ambitious renewable energy and carbon reduction targets.
"The proposed European Offshore Wind Deployment Centre will give the industry the ability to test and demonstrate new technologies in order to accelerate its growth.
"It secures Aberdeen's place as the energy capital of Europe."
He added: "The EOWDC is the first offshore wind farm to pass through the new approvals process for offshore wind development and as such is a test case for the industry. The diligence and expertise of officials in Marine Scotland, among the statutory consultees and other interests should give confidence to the offshore wind industry and its supply chain that planning matters should not be a barrier to investment in Scotland."
EOWDC project spokesman Iain Todd said: "The Scottish government's most welcome approval for the EOWDC is extremely positive news for both Scotland and the UK's offshore wind industry as it helps position Scotland, the UK and Europe at the global vanguard of the sector.
"The project partners will assess the consent and associated conditions as part of the planning and programming for the next phase of the scheme."
Alex Salmond, the MSP for Aberdeenshire East, said: "I welcome this decision.
"The north east of Scotland is a world energy capital and in order to continue to prosper we have to be at the forefront of all forms of energy research - marine renewables, as well as oil and gas."
Dr Sam Gardner, senior climate change policy officer at WWF Scotland, said: "Giving the go-ahead to this offshore wind test centre is the right decision, demonstrating that no amount of bluster from US billionaires such as Donald Trump will hold Scotland back from becoming a cleaner, greener, job-creating nation."
Robert Collier, chief executive of Aberdeen and Grampian Chamber of Commerce, said: "The north east is already established as a global hub for the oil and gas industry.
"The development of the EOWDC will enable the region to extend its reach with regard to offshore wind, further enhancing our renewable energy expertise and drawing upon our strong engineering and technology skills base.
"This is an important development for the region."
Consent for a wind farm off Aberdeen which is opposed by Donald Trump has been granted by the Scottish government, angering the US tycoon.