No, normally it is not possible to get seeds from store bought bitter melons. The fruits are immature and hence the seeds are not fully developed. When the fruits are mature they turn red and the seeds will be ready. You'll need to either buy some seeds or get some from a friend.

Good luck.

:-( Thank you. Would my method work, putting out the seed flats in the mornings in full sun and bringing them inside at nights? Or would i kill them?

It depends on the zone you are in. I live in Atlanta (guess zone 7B and 8) and I put the seed outside in the garden first week of May and I have the plants coming up. You may not need to start the seed indoors if the temperature outside does not fall below 50F or does not have consistent freezing temperatures. Bittermelon is a very hardy plant. You should not have problems with germination unless the seed itself is very old or like you wrote from an unripe bittermelon.

True, just don't let them dry out.

Here's a few pics of mine.
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I successfully sprouted gourd and bitter melon seeds in a sandwich bag. Soak seeds for 24-hrs in water, then wash a and wrap in paper towel wet in warm water. seal, and put in a dark warm place. under the oven. it takes about a week to sprout.

I had no luck with snake guard, so I am going to try to soak longer and remove the shell and try it that way

They are real stubborn plants...
You might have ended up with wrong seeds.
Buy them online they are not that expensive.
I had one plant last year and it did well.
Unknown to me a melon had ripened and I pulled it out after
its seeds fell down on the ground.
Amazingly I got a surprise plant his year from that fallen seed...(thanks to the mild winter we had..)

I use the paper towel and plate method to germinate my bitter melon seeds.

First I make sure that everything was sanitized first to reduce the possibility of mold. (i.e. plate from the dishwasher, clean paper towel, etc.) If I am especially concerned about mold, then I use a tiny dab of chlorine bleach in the water that I'll use for moistening the seeds.

I put a small amount of water (I use reverse osmosis purified water) into the plate and a few drops of chlorine bleach if concerned about mold. Then I put a folded paper towel onto the water. The objective is to have the paper towel be moist, but NOT dripping wet.

Then I carefully arrange my bitter melon seeds on the paper towel so they are not touching, then fold the top of the paper towel over them. Then the whole plate is covered with plastic wrap, and the plate is put on a shelf in the house out of any direct light. Temperature in the house is 76 to 78 degrees F.

After the first couple of days, the plate is checked every day for germination and to make sure that the paper towel isn't dry. It needs to be kept barely moist.

If a root is starting to emerge from a seed, that seed gets transplanted into a wide squat seed cell--the ones I use for bitter melon seeds are 2 1/4" x 2 1/4" x 2 1/2" deep. The seed cells are kept in a sheltered spot until the seedlings emerge from the potting mix and then moved outdoors to grow out.