International Business Machines Corp. won’t get a full Federal Circuit rehearing of a ruling that cleared online booking sites Kayak, OpenTable, and Priceline of patent infringement.

The Booking Holdings Inc. websites and mobile applications don’t infringe U.S. Patent No. 7,072,849, which involves methods for presenting advertising over a computer network, the U.S. Court of Appeals for the Federal Circuit ruled in May.

In requesting a full-court hearing, IBM counsel Karim Z. Oussayef, a partner at Desmarais LLP, said the case “requires an answer to a precedent-setting question of exceptional importance: whether an entity performs a method step by causing third-party...