NO. The car starts off of the battery, it runs off of the alternator. This has been true since the beginning of time. Back in the olden days, Grandpa cranked the engine over by hand, then a magneto took over keeping the engine going -- just like a lawn mower. Then cars started getting electrical stuff -- lights and such -- and a generator was installed. Grandpa still cranked the car, but the electrical loads were sustained by the generator. Somebody got the bright idea that an engine could be started with an electric motor, but it took an electrical charge to make that happen. Enter the battery. The generator still provided for the electrical needs of the engine, and it also replenished the battery. Eventually the generator was replaced by an alternator -- the generator produces DC while the alternator produces AC that a diode pack changes into DC -- but the idea that the battery starts the engine while the alternator sustains it remains the same.

It is not a good idea in modern systems to unplug the battery while the engine is running, but in a car from before the days of computer controls it was common practice to start a car, then take the battery out and move it to another car so it could be started. Jumper cables make this much easier, but sometimes one might not have cables but for some reason was able to harvest a battery from another car.

The point is, the engine runs because of the alternator, it starts because of the battery. If the battery is gone, starting is lost, if the alternator is lost then engine operations will be lose when the battery goes flat. This can take varying amounts of time depending on the loads and the condition of the battery.