"He didn't want to" is probably not a super-satisfying answer, but it's the long and short of it. I'll give you two quotes from him about the matter, the first from 1886 after his stint as Secretary of War:

God willing, I will never again be in the jaws of that damning hyena, the public at large

And one from 1887, during the ramp-up to the 1888 campaign:

To take any office at all would be a great sacrifice on my business interests here in Chicago; and the Vice Presidency is not an office of such importance that I could afford to think of such a thing.

So he had his reasons, and he did also refer to the Presidency as a "gilded prison." He was very popular, though, and would probably have at least gotten nominated by the Republicans.

Interestingly, he was associated with three presidential assassinations, which led to a reluctance on his part to be in the presence of the president. His father was the first, and he was a witness to the James Garfield assassination in 1881. He was also at the Pan-Am Expo in Buffalo when McKinley was assassinated in 1901, but he did not see the shooting. This may have contributed to his not wanting to run for president.