The order was issued Thursday.
Simmons' attorneys had asked the justices to stay his execution for two weeks because they say more time is needed for two mental health evaluations and an appeal based on their results.
On June 5, the court set the execution date for Simmons, but granted requests for evaluations by a forensic psychologist and a neuropsychologist. His lawyers filed another motion Monday, contending there isn't enough time before the scheduled execution to complete the evaluations and, if warranted, to file an appeal based on the results.
The Mississippi attorney general's office responded Wednesday in a court filing that said the request is a delay tactic and argued the court should rescind deny a stay of execution.
Simmons' lawyers argued that he suffers from a mental illness and was abused as a child. If evaluations support the claims, lawyers want the court to order Simmons resentenced, giving him a chance to avoid execution.
The attorney general's office filing said Simmons' sanity "is not in question" and that if he's allowed to get the evaluations then other death row inmates will ask for the same type of tests to delay their executions.
In a response filed Wednesday, Simmons' attorneys argued his due process rights would be violated if sufficient time isn't permitted for the evaluations.