MEXICO CITY—Free-speech advocates are challenging a ruling against Google Mexico that they say would allow politicians and business moguls to abuse the so-called right to be forgotten by wiping out Internet links that cast them in a negative light.

The Google Inc. unit and local digital-rights activists are fighting in court to overturn a recent ruling by Mexico’s Federal Institute for the Access to Information, or IFAI. In late January, the institute came down on the side of a transportation scion who wants three links removed...