Alphabet Inc. can’t pause a patent infringement and trade secrets lawsuit involving its Project Loon aerial internet effort while it tries to kill a related patent at the Patent and Trademark Office.

Space Data Corporation sued Alphabet and its Google LLC and Loon LLC subsidiaries, claiming they infringed four of its patents and illegally used trade secrets with their Project Loon, an effort to use high-altitude balloons in the stratosphere to provide wireless services in remote areas. Space Data Corporation also alleged Google illegally used its confidential information.

A spokesperson for Loon said the company “doesn’t believe Space Data Corporation’s...