The answer to this is YES. You can successfully grow a nectarine seedling from the seed of a nectarine fruit. However, it is worth noting that, like many other fruits, it is very unlikely that the resultant tree will bear fruit that resembles that of the tree that it came from. This is because the parent tree that produced the delicious store bought fruit was a hybrid, produced by budding or grafting, and not grown from a seed. This allows growers to combine desirable characteristics, but the seeds will not breed true to type. In addition, nectarine seeds need to be cold treated to break their dormancy before they will germinate.
If you don't mind that and would like to try growing your own nectarine seedling (I think its really fun and am currently growing avocado seeds) then it is relatively straightforward. Here is what you will need to do:
How to grow a nectarine from seed
1. Remove the nectarine pit from the fruit and place it in a bowl of water to soak off any remaining pulp. If the pit floats then its not viable and you will need to try again with another pit. Air dry the seeds on the counter top for a 1 or 2 days.
2. Remove the seeds from the pit by cracking it open - try using a pair of nutcrackers to crack it open. Once removed, soak the seeds in a bowl of water overnight
3. Straitify (cold treat) the seeds. This process essentially simulates winter conditions and breaks the dormancy of the seed. Fill a lidded container half full with some moist potting soil and place the seeds on top of the soil. Close the lid of the container and place in the fridge. Leave for several weeks, make sure that the soil is kept slightly moist and does not dry out. After about a month, the seeds should begin to germinate
4. Plant each germinated seed in a small pot (about 6" diameter) filled with good quality potting soil. Plant the seed 0.5-1.0" deep in the center of the pot. Keep the soil moist but not wet and continue to grow the seedlings.
5. Transplant the seedlings into the garden or continue to grow as a container plant.
Hope this helps - I kind of want to try this too! Good luck!
Nectarine pits need cracked open to remove the seeds inside