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Time Warner Cable is making a federal case out of its dispute with Viacom over the right to stream shows to the iPad.

The cable-Internet giant, which serves much of New York City and about 13 million people in the US, asked a judge yesterday to give it legal cover to stream Viacom’s content to Apple’s popular tablet after the entertainment giant threatened to file a lawsuit to block them from doing so.

Unbowed, Viacom answered TWC’s legal action with one of its own — suing to block any such streaming and award damages for violating its copyright materials.

At the heart of the case is TWC’s new app for iPads that allows its subscribers to view the same content they have access to through their televisions. The app, launched March 15, was downloaded by more than 300,000 people. Viewers can only access the app while at home.

The No. 2 cable company’s contract with Viacom does not “limit the types of video display devices upon which TWC subscribers may view programming provided through TWC’s cable system,” according to the cable company’s lawsuit filed in Manhattan federal court.

After TWC launched its app, Viacom was quick to object and have its channels, including MTV, Comedy Central, BET and Nickelodeon, pulled from the service. Since then the channels have not streamed to iPads, but the company’s suit seeks to change that.

In its lawsuit, TWC likened streaming to the iPad to delivering content to different types of televisions, which have evolved over the years from analog cathode-ray tubes to digital flat screens.

At stake is more than the iPad as the next-generation of Smart TVs will continue to blur the line between what is delivered through the Internet and what is delivered through cable.

Viacom, in its lawsuit, said it was concerned with protecting its content that it already streams through other online channels, such as Hulu and its own apps and Web sites.

Viacom said it is also concerned with product quality and the ability to measure the audiences of its programs.