An attorney for Miller sought the preliminary injunction Thursday as part of a previous lawsuit in which he has challenged write-in ballots for Murkowski.
A judge last week denied Miller's request to stop elections officials from counting write-in ballots that stray from the exact spelling of a candidate's name.
Attorney Tom Van Flein said in his filing Thursday that Miller was revising his injunction request because elections officials are nearly finished with the vote count.
Murkowski has a lead of about 10,400 votes. Miller had challenged 8,153 of the ballots counted for Murkowski, and would still be behind even if he won every challenge.