Short answer. Generally no. Once the period to appeal expires with no appeal perfected, Rule 39, Section 1 makes execution issue as a matter of right, on motion. The court's duty at that point is not discretionary — but you must still file the motion, and after an appeal you must attach certified copies.
What the law says
Execution shall issue as a matter of right, on motion, upon a judgment or order that disposes of the action or proceeding upon the expiration of the period to appeal therefrom if no appeal has been duly perfected.
Rule 39, Section 1 — Execution upon judgments or final orders. Read the full provision →
Execution as a matter of right
The section's opening line is the whole answer for most cases: Execution shall issue as a matter of right, on motion, upon a judgment or order that disposes of the action or proceeding upon the expiration of the period to appeal therefrom if no appeal has been duly perfected. Two conditions, both about time: the judgment must dispose of the action, and the appeal period must have lapsed without an appeal being perfected. When both are met, the prevailing party is entitled to the writ — the language is shall, not may.
You must still ask
A final judgment does not enforce itself. The right arises on motion, so the judgment obligee must file one before any writ issues; the court does not act on its own. Where the case went up on appeal and was finally resolved, the section adds paperwork: execution may forthwith be applied for in the court of origin, on motion of the judgment obligee, submitting certified true copies of the judgment or judgments or final order or orders sought to be enforced and of the entry thereof, with notice to the adverse party. Missing documents, not judicial reluctance, are the usual cause of delay.
When the appellate court steps in
The final sentence covers the situation where the court of origin does not move after an appeal: The appellate court may, on motion in the same case, when the interest of justice so requires, direct the court of origin to issue the writ of execution. So a winning party who meets resistance below has somewhere to go — the same appellate case remains a live avenue for compelling issuance. Note the shape of the provision: everything in it is about how execution happens, and nothing in it gives the trial court a ground for refusing a writ that the first sentence commands.
What to check before you move
Confirm the judgment is genuinely final: the period to appeal must have expired with no appeal duly perfected, or the appeal must have been finally resolved and the entry of judgment made. Secure the certified true copies the section requires before filing, and give notice to the adverse party. If the court still does not act, the record you built — motion, certified copies, notice — is what an appellate court will look at when asked to direct issuance. The section presumes a diligent movant; a winner who never files the motion has not been refused anything.