Is it permissible for someone who has performed the tawaf in Umrah to delay the sa'y for two days, while entering the Grand Mosque and praying in it with the intention of greeting the mosque during these two days, or was it obligatory for him to perform tawaf every time he entered the mosque?

Delaying the Sa'y after the Tawaf without a valid excuse is reprehensible (makrooh). It is sunnah to repeat the Tawaf along with the Sa'y if the separation is prolonged; otherwise, nothing is required. The greeting for the Sacred Mosque for someone entering for the first time is the Tawaf, unless there is a fear of missing an obligatory prayer, its congregational prayer, or an emphasized sunnah. As for someone who enters the Mosque for a purpose other than Tawaf, their greeting is two rak'ahs, like any other mosque.