Sherman was never president of the United States, though a good many Republicans in the 1884 thought he would have made a fine candidate for that office. He had retired the year before as commanding general of the United States Army and was still a household name -- and at least in the North, admired for his part in the war.
Sherman was having none of it. Perhaps it was because he'd seen how vexed his former commander US Grant had been by the presidency. More likely, a visceral dislike of politics kicked in at the idea of running for president. In any case, he answered the suggestion that he run with the definitive statement of a non-candidate, one that echoes down the century and more since then: "If drafted, I will not run; if nominated, I will not accept; if elected, I will not serve."
Instead, he lived in retirement in New York City until his death, attending the theater, painting, making dinner speeches when he was a mind to. As for the election of 1884, James G. Blaine, former senator and secretary of state, was the Republican nominee. President Arthur, who had succeeded to the office after the assassination of James Garfield in 1881, chose not to seek the nomination either, probably because of illness. The Democrats nominated Grover Cleveland, the governor of New York, who won in a tight race against Blaine.