"launch a lawsuit" is correct and usable in written English.
You can use it when you want to express the concept of initiating legal action against someone or some entity. For example, "The company decided to launch a lawsuit against the former CEO for misappropriation of funds."
After being told of Banon's intention to launch a lawsuit, Strauss-Kahn's Paris lawyers said he would sue for slander.
He set up a private fund to build the Centre of the Tokyo Raids and War Damage, and helped launch a lawsuit for compensation.
September 2014 Investors launch a lawsuit against SeaWorld, claiming the company misled them about the impact the film had on attendance of its parks.
The warehouse has been used for graffiti for more than two decades, but the property owners have plans to demolish it, prompting the artists to launch a lawsuit to save the space.
Filmed in French, with a French cast (with the exception of a cameo by Jodie Foster) and a French crew, the financial support of Warner Brothers has prompted French film-makers' unions to launch a lawsuit over whether the film is eligible for the public subsidies that French films usually get.
Faced with the near-certain destruction of their homeland – part of the US Commonwealth of the Northern Marianas – dozens of former residents have joined forces with environmental campaigners to launch a lawsuit they hope will expose the folly of the Pentagon's plans to transform Pågan and Tinian, an inhabited island 200 miles to the south, into simulated theatres of war.
Sony launched a lawsuit days later, saying Mr Hotz was encouraging the use of pirated games.
The four men are launching a lawsuit, which was filed with a New York court on Monday.
Regulators in Europe and Asia are investigating, and AMD has launched a lawsuit against Intel in America.
Another of the sacked three launched a lawsuit against the carmaker.
Now, two years later, comes news that the C.I.A. has launched a lawsuit against him.
