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Birth of an Island
slashmojo writes that while some islands are sinking, last August another rose from the ocean, formed by volcanic activity and caught in the act by a passing yacht. From the article: "What looked like a brown stain on the South Pacific turned out to be a spectacular drift of floating pumice stones stretching more than 16 km — and an indication an island was being born nearby... 'We are getting emails from volcanologists saying this is so rare.'" Here is the blog post of the yachtsman who photographed the nascent island.
Please look at http://en.wikibooks.org/wiki/Making_an_Island for some further information and sources.
In the early 1970's, there was a Las Vegas developer who ended up going to an atoll that was technically in unclaimed international waters and "built" an island by dumping extra material on this group of submerged rocks to the point that there was a portion that stayed above water during high tides, technically new territory just as you have suggested.
BTW, this was also near the Tongan islands, so this is also relevant in this situation.
What happened afterward was that a group of Tongan soldiers "invaded" the newly formed island and asserted sovereignty by "occupying" the island in the name of Tonga. Instead of formenting an international incident, the developer relented and gave up his attempt to build his own South Pacific version of Monaco.
I'm not sure what would have happened if this developer had his own "army" that would have defended the island, but it certainly seems like Tonga would consider it justifications for going to war if it happened near one of their islands. I'm curious what the Tongan government may have to say about this new island in their general domain.