There is no precedent that Trump can sign an executive order eliminating his opponent from the race.
In his speech, he stepped up attacks against his opponent, Democratic presidential nominee, Joe Biden, by accusing him of taking drugs and calling for a drug test. He called him the "dumbest person" and the worst candidate in the history of presidential politics. At one point in the speech, he even threatened of disqualifying him from running for president by passing an executive order. He did not provide evidence for his claim about Biden taking drugs, nor is there precedent that he can sign an executive order to eliminate his opponent from being elected. All executive orders should be constitutional, and because the rules for who qualifies as a candidate for election and how are listed in the constitution, and Biden meets all criteria, no executive order can bar an otherwise qualified candidate from the presidency.
Biden and Trump are to have their first 90-minute debate on Sept. 29, 2020, in Cleveland.