H-1B Stamping Dates - Consulate open COVID-19?

I travelled India on the 13th of March and could not get my visa stamped due to the appointment cancellation. I just heard that the USCIS would open only on June 3rd, does it mean that visa consulates won’t open till that date and my visa stamping won’t be possible before that date.

USCIS has nothing to do with US Consulates. Both are different.
Every country has their own requirements for opening US embassies, so you need to check with the consulate in your country for the latest info on when they will open the consulate.