The man, 42, and woman, 30, were taken to Salisbury District Hospital after emergency services were called to the restaurant at 6.45pm.
Due to recent events in the city, scientific tests were undertaken to establish if they had come into contact with the nerve agent novichok.
The tests concluded this was not the case and their illnesses are not being linked to poisonings in Salisbury and Amesbury.
The former Russian spy Sergei Skripal and his daughter Yulia were targeted with novichok in March.
British intelligence has blamed the Kremlin for the attack.
Dawn Sturgess from Amesbury died after finding a perfume bottle containing novichok in June.
Enquiries into what caused the pair in Prezzo to fall ill are still ongoing, but the case is not being treated as suspicious at this stage.