No, your apricot probably came from a cultivar grafted onto a non-related rootstock so the seed will not "produce a rootstock true to the original tree's rootstock". The seed would have absolutely no relation to the rootstock as they both came from different parentage.
If the tree was cross pollinated to produce the fruit it most likely wouldn't even grow to produce fruit that resembles the fruit the seed came from either. This is where 'chance seedlings' come from - seedlings that are allowed to grow and turn out to display desirable characteristics not yet exhibited by other cultivars.
I have no experience with stone fruits, but I have read that apples carry far more genetic diversity than anything else we cultivate, which is why you're almost guaranteed to get something you'd rather not eat if you grow from seed. Probably any other fruit is more likely to produce something closer to what you expect, but "more likely" may not be likely enough. The problem with this kind of experimentation is that it takes so long to know the result.
That's precisely what I understood, Marsha. I was under the impression that you had to plant literally 100's if not 1000's of apple seeds to get a result that wasn't a spitter (or at least that's what I've been telling people and sounding very impressively knowlegeable about the complexity of the apple genome, as one portentously tends to). And then I very recently read a very experienced hand saying "plant those apple seeds out...50% won't work - they'll be bi-annual or dwarfing or disease-susceptible, diseased etc. - but half will turn out just fine. "At least as good as the parent," he said. Has anybody hereabouts actually put this to the test?
From what I’ve read modern breading consists of controlled crosses with known cultivars, generally a very popular model and something less known but more likely to provid a genetic advantage long bread out of the popular one.
Our Pome News had a good segment on ‘spitters’ while further describing the adventures of “Johnny Appleseed.†Seems he was promoting cider apples with his incessant planting of seedlings and really didn’t care what they tasted like, as long as they produced drink!
As I yearly press and dump the pulp of seasonal batches of fresh apple juice, and have acquired some top notch cultivars - I’ve often wondered just ‘what’ I’m tilling under when grinding up seedlings by the dozen. It’s somewhat painful… but given the odds I’ve heard of one in six thousand being ‘superior’ to its parents, I haven’t the time or room for that action 
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Yes, that 1 in 6K number was what I had taken as received wisdom. But here's the original quote, a report in the Autumn 2003 Alaska Pioneer Fruit Growers Newsletter, on a presentation by one of Canada's foremost fruit enthusiasts, Bernie Nikolai, out of Edmonton, Alberta: "Bernie encouraged us to plant our own seeds and develop our own varieties, not only for rootstock but for apple production. Take seeds from locally grown apples, rinse and dry them overnight, then plant them in the ground about 2 feet apart in September and wait. About half will be biannual or dwarfing or prone to fireblight etc. but 50% to 70% will turn out to be a good apple - at least as good as the parent. Production only takes about 5 to 8 years, so if you have the time and space, plant a few."
That rather raised an eyebrow.
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I am allowing some rootstock to grow apples. That is mainly because most apples from rootstock or random, will be somewhat bitter and not good tasting. This is terrible for fresh eating, but great as part of a cider blend. I look forward to tasting some and adding them to cider. Bitter is good for part of cider. I still don't want it to be a large portion of my apple harvest.
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