No, because the part of the plant that is being tissue cultured is the immature embryo. After pollination, the flowers of seedless grapes still produce an embryo, but the embryo stops developing when it is very small, and there is no fully formed seed (a process called stenospermocarpy). To grow the offspring of a cross, breeders have to dissect the immature fruit (the flower's ovary) and remove the immature embryo. The embryo is then transferred to tissue culture media where it is grown into a plant with roots and leaves that can eventually be transferred to soil.

Are you certain the grapes are cloned using the embryos? Or are they propagated from cuttings?

In the breeding process, I am absolutely sure that during the breeding process they propagated using immature embryos. "Cloning" by definition implies vegetative propagation, and I'm sure when they finished the breeding process that is how they have propagated this cultivar.

If you are interested in more of the details, there is a wikipedia article on cotton candy grapes, If you look in the references, there are several articles with more on how the variety was produced.