So did the U.S. use the second -- and likely false -- attack as an excuse to invade Vietnam?

I, like my grandfather, would say yes.

I did some research and stumbled on to some audio recordings between President Johnson and Defense Secretary, Robert S. McNamara. There is a conversation, from the morning of August 4, 1964, which Johnson and McNamara discuss retaliatory action against the North Vietnamese. President Johnson says to McNamara, "(you) should pull one of these things that you've...been doing...on one of their bridges or something."

According to this web page, which is a part of the National Security Archive, "This is a clear reference to the OPLAN-34A raids, confusion about which had been a factor in the initial Tonkin Gulf engagement on August 2. Here LBJ suggests a measure that would actually increase Hanoi's incentives to fight" (LBJ Tapes).

This conversation basically says that the U.S. is ready to attack North Vietnam.

Funny thing though: this conversation occurred before Johnson and McNamara had gotten word about the alleged second attack!

It seems that the U.S. was prepared to go to war anyway, and this second attack gave the perfect excuse to heighten military involvement in Southeast Asia.