How did the United States respond when North Vietnam attacked a US warship and sent troops to South Vietnam in the early 1960s?

The United States declined to assist South Vietnam against the North.

US leaders authorized South Vietnam to invade North Vietnam.

The president did not sign the Gulf of Tonkin Resolution.

Congress authorized the president to escalate the conflict.

First of all, the chronology of the question is slightly incorrect. The US sent its first troops and military advisors in 1962, under the Kennedy Administration. Second, it has been proven that there was actually no initial attack from the North Vietnamese on a US Warship. According to an NSA report of 2005:

"At 1500G, Captain Herrick (commander of Maddox) ordered Ogier's gun crews to open fire if the boats approached within ten thousand yards. At about 1505G, Maddox fired three rounds to warn off the communist [North Vietnamese] boats. This initial action was never reported by the Johnson administration, which insisted that the Vietnamese boats fired first."

Nevertheless, the correct answer is: Congress authorized the president to escalate the conflict.

Indeed, President Johnson ordered American bombings on North Vietnamese bases to take place immediately. HE then authorized the sending of additional troops to Vietnam. In 1971 the Tonking Gulf Resolution was repealed under President Richard Nixon.