No, airbags deploying does not automatically make a car a total loss. If a vehicle's airbags deploy and the cost of replacing them is more than the total loss threshold for your state, it would be declared a total loss. See the total loss threshold by state to check for yours.
Airbags deploying is like any other damage to a vehicle; if it is more economical to total out the vehicle than to repair it, then an insurance company will do so.
Due to the high price of new airbag components and installation costs, many vehicles with a low actual cash value (ACV) will be considered a total loss if the airbags deploy. On the flip side, if the vehicle has a high ACV, the cost of new airbags being installed will not likely cause the vehicle to be declared a total loss.