He did tries to run out his competition but he never could completely eliminate them. At one time, Standard Oil only fleeting hold 91% and 85% of final sale. So there were still competitors at the hieght of his empire.
However, by the time the federal government decided to use antitrust charges on Standard Oil, its share of the market dropped tremedously.
In any case, the government is itself responsible for creating legal monopolies such as patents and copyright. It was also supposed to break up supposed monopolies, which may or may not be true. With such conflict of interest and the law of unintended consequence in mind, I am very skeptical about such antitrust claims.
Don't take my word for it though. Do your own research.