Mr Puigdemont is living in self-imposed exile in Belgium and is wanted by Spanish authorities for leading a failed independence bid in 2017.
Pre-election polls had predicted a bigger victory for the PP and the possibility for it to form a coalition with Vox.
PP and Vox have already teamed up to govern in dozens of regions and cities since local elections in May.
Vox, founded in 2013, has proposed the expulsion of illegal migrants and a naval blockade to stop them arriving, and the closure of mosques promoting “radical Islam or jihad”.