I'm looking at the new Warlock eldritch invocations of XGTE, and the invocation Gift of the Depths gained my curiosity.

The eldritch invocation Gift of the Depths states that you can breathe underwater (emphasis mine):

You can breathe underwater, and you gain a swimming speed equal to your walking speed.

If you can breathe underwater, can you also rest underwater?

Yes, it is possible (depending on your circumstances and DM)

Nothing in the rules prevents this

There is nothing in the rules for resting that prescribes the environment that you can rest in. There are only certain things that interrupt rest according to the rules:

If the rest is interrupted by a period of strenuous activity - at least 1 hour of walking, fighting, casting spells, or similar adventuring activity - the characters must begin the rest again to gain any benefit from it.

Thus, if it is safe and you can breathe you should be able to rest according to the rules alone.

However, you DM might have other concerns about resting and sleeping under water that are not covered by the rules.

Environmental conditions might make sleep tricky for most races¹

Short rests just involve a short period of respite. A long rest involves sleep though (for most races). Unlike rest, there are no rules about sleeping or what can disturb it so a DM must make some rulings. Breathing is obviously a good place to start when choosing a sleeping location, but there are also other considerations. How important these considerations are to you and your particular circumstances will be up to your DM.

Temperature: For example, water is often much cooler than body temperature. Trying to rest for long periods of time in cold water might be dangerous (hypothermia) and uncomfortable and might be ruled to prevent sleep for those reasons.

Moving water/buoyancy: In water, a human-like body is buoyant and trying to sleep on a surface under water could prove tricky you might have to tie yourself down to actually sleep on a surface. Also, not all water is still and could cause a lot of shifting (even dangerous amounts) that could affect sleep. Watch out for tides and currents! Think of it this way: laying on a pool on a inflatable raft is very easy to sleep on. You put that raft in a white water rapids and I'll bet sleep doesn't come as easy!

Most, if not all, of these seem overcomable in one way or another (especially with magic) and, over time, a character could learn to rest comfortably under water. And as a DM it seems like a fun thing to help a player do if they want and if it is relevant. A DM can may of course rule that, depending on the exact circumstances, that it is not possible to rest.

^{1 - This answer assumes a human-like, land-dwelling physiology obviously. Other races might have quirks that make it easier or harder to sleep underwater. For example, something like a water-dwelling triton would have no issue sleeping under water.}

It is possible

It's an environment that you can breathe and be in so there really isn't any general reason why couldn't rest there like you could on dry land.

Environment may present issues

However, there may be circumstances that a DM could employ that could prevent resting that aren't as common on land such as currents preventing you from staying in one place.

Rubiksmoose also covers another aspect: maintaining a comfortable temperature. This may be table variable, but it's something else for a DM to consider when determining if it's viable. This also goes for issues regarding equipment depending on how closely those types of things are tracked.

Ultimately, the general answer is Yes, but there are enough environmental variables that a DM could say No.

You can rest underwater if you can breathe underwater

You can rest anywhere so long as you don't die whilst doing so (or are interrupted by combat or other strenuous activity):

If the rest is interrupted by a period of strenuous activity - at least 1 hour of walking, fighting, casting spells, or similar adventuring activity - the characters must begin the rest again to gain any benefit from it. (PHB)

The only reason that a typical adventurer wouldn't be able to rest underwater, as it stands, is because they would drown before they completed their rest. So if you gain the ability to breathe underwater, there's nothing stopping you from resting underwater RAW. However, as Rubiksmoose points out, there may be environmental concerns that a DM may wish to take into consideration.

That is fine. Breathing underwater and a swim speed are more than enough to allow a character to rest underwater.

Sea elves (Unearthed Arcana) have exactly that ability as a subrace trait. They can breathe underwater and have a swim speed. They live underwater and of course they rest underwater.