If one day we have the biggest earthquake/rift under Pacific Ocean and somehow create a new continent, what is the legal process of claiming an ownership?

There's no real treaty for this.

It's not entirely hypothetical. We are in a time where discovering things of great value to multiple parties in International Waters is plausible. Undersea mining... oil/nat gas fields that could never be mapped or drilled before are becoming more plausible to exploit.

Also similar, the arctic was pointless to claim before, so no one argued claims to it. Now, it's got some value and promise of ever-increasing value and lots of countries are arguing ownership.

Space exploration, too.

https://en.wikipedia.org/wiki/Law_of_the_Sea_Convention[1]

Exclusive economic zones extend 200NM from the low-water mark on the shore.

In International Waters, the seabed floor is "controlled" by International Seabed Authority[2] , but it's kind of a joke. It's a small body which meets infrequently and doesn't carry a lot of authority. The US wants the ISA dropped.

https://en.wikipedia.org/wiki/Common_heritage_of_mankind[3]

Is the principle that says the Moon cannot be claimed or exploited by any one country. That really wouldn't hold up if something of significant commercial value showed up there.

Common Heritage of Mankind definitely wouldn't hold if a new, usable continent showed up.

Basically a clusterfuck. I expect the following:

1. We made the first landing there and own the whole thing
2. No, a fisherman of your nationality, not your government, landed there first. Actually we landed there 2 days before that you just didn't hear about it. Basically it will be a claim no one respects, except for the person claiming the flag-planting.
3. Country A is closest point to a shore, we own the whole thing.
4. No, Country B owns the half facing its country
5. We should all divide it up to ALL COUNTRIES EQUALLY based on existing country size (hurray Russia)
6. No, we should divide it up based on population (hurray China)
7. Most countries: well when we say "divide it up", we don't actually want to USE it and have territory next to you. We'll SELL our claim to another country. So, like, the US or Russia or China should just give us a boatload of $$$ to ALL these little guys who never set foot on the place, just because they exist.
8. Some minority offended in the past: Native Americans should occupy it since you took our country. It's a new Jewish homeland, it's Zion but we never really knew what that meant until now.

On point #7... let's say someone finds a way to get a valuable resource off the Moon. But only one country has the tech to TAKE it. Maybe it's such a specific product only that country can even use it. Still, other countries would demand payment for their rights to it. Not necessarily cash, they would more likely use it as a political pawn- "if you want to mine the lunar regolith for Helium-III to make fuel to go to Mars, then China demands you recognize our rightful claim to the Diaoyu Islands and we will not stop you." "But you don't have any interest or ability to mine that Helium-III!" "And you have no interest in colonizing the Diaoyu Islands, nor does Japan. So back China's claim and that works out nicely." Then Japan backs off and says "well we WERE going to greenlight your Moon-mining plan because it made sense, but now we see that's a mistake. No, WE will block it unless Diaoyu/Senkaku Islands are recognized as JAPAN'S claim!"