Short answer
No. A hen’s reproductive cycle produces and lays eggs without a rooster. A rooster is required only for fertilisation and the possibility of developing a chick.
Laying is part of the hen’s cycle
A yolk is released from the ovary and moves through the oviduct, where albumen, membranes and shell are added. Hormones, daylight, breed, age, nutrition and health influence this cycle. Mating is not required for it to occur.
What fertilisation changes
If sperm are present, fertilisation can occur shortly after ovulation before the shell forms. A fertile egg still looks and cooks much like an ordinary table egg when freshly laid. Development requires suitable incubation after laying.
Why supermarket eggs do not hatch
Commercial table-egg flocks generally contain no roosters, making the eggs infertile. Even a fertile farm egg needs sustained warmth, humidity and turning to develop. Normal collection, transport and refrigeration interrupt those conditions.