It would quickly* rupture.
Five-hundred pounds of water is only about 8 cubic feet in volume, so the bag's weight capacity would be overloaded after merely an eighth of its volume capacity was reached — assuming it started empty, as any other items in it would help it reach its weight limit earlier. It would then rupture, scattering its (damp or soaked) contents across the Astral sea.
An objection might be raised: doesn't the bag only store things specifically put into it? No, the owner is not so in control. There's no verbiage limiting access to only intentionally stored items, so an open bag allows anything to pass into it. This means that opening it underwater would result in the water pouring in.
The space inside the bag is described as a continuous finite volume, not a set of pocket-spaces for each individual item put in it; therefore, putting/letting water into it would get water on any other items sitting inside.
* How quickly depends on the local water pressure, but just “quickly” is plenty of precision for our purposes.
Yes, but that'll destroy both and create other havoc.
Placing a bag of holding into an extradimensional space created by a Heward's handy haversack, portable hole, or similar item instantly destroys both items and opens a gate to the Astral plane.... (DMG p.154)
So the question, then, is whether a bag of holding, itself, creates such a space and is thus a similar item.
(And it's tempting to think that it's not: the bag's own description doesn't say that the interior is extradimensional, just that it's larger than the outside. For comparison, both the haversack and the hole do explicitly say early in their descriptions that they create extradimensional spaces.)
But the bag does create an extradimensional space. We know this from the descriptions of the handy haversack and portable hole, each of which list the bag of holding as creating one of these dangerously-incompatible spaces.
See DMG pp. 153-4, 174, and 186-7 for all descriptions.
Later, Mearls agrees. Remember, his rulings are not authoritative, but rather are just the word of one very well-informed GM.
Best Answer
According to the item properties in the DMG, no, it can only be opened from the outside. Specifically, items inside require an action to be retrieved.
(To satisfy those who have asked: the word retrieve, as used here, is a transitive verb with an implied indirect object, usually the same as the subject. The direct and indirect objects of a transitive verb can very rarely, if ever, be the same in English. Therefore, no, something can't retrieve itself, because that doesn't make sense.)
However, YOUR bag of holding does not have to be the bag of holding presented in the book. If you say that it can happen in your campaign, then it can. Never let minor mechanical details get in the way of the story (being mindful of your players, of course).