No, the reason is because of the mascoms - the concentrations of mass on the Moon.

​As satellites orbit the Moon their orbits are tugged one way and then the other by the Mascons, and these keep changing the shape of the orbits. This doesn’t make them spiral down - it's not a tidal effect. Instead, it varies their ellipticity, sometimes more, and sometimes less elliptical. Eventually, in most orbits around the Moon the orbits become so elliptical they intersect the surface of the Moon and the moonlet crashes.

"FROZEN ORBITS" OF THE MOON

This was first discovered in April 24, 1972. The Apollo 16 astronauts tried to put a small satellite called PFS-2 into lunar orbit to orbit every 2 hours. An earlier satellite PFS-1 was released by Apollo 15 and had been orbiting the Moon just fine for eight months. And they were in similar orbits originally, ranging from 55 to 76 miles above the surface.

But - to their surprise, PFS-2 rapidly changed the shape of its orbit. Within two and a half weeks, it was swooping down to within 6 miles of the surface. After a while it backed away again to 30 miles from the surface, but eventually, only 35 days after it was released, it hit the Moon.

PSF-1 lasted for a year and a half, before it hit the Moon. And there are a few "frozen orbits" where a spacecraft can orbit the Moon indefinitely. So those are good for mission planners who want their satellites to orbit the Moon for a long time without using a lot of fuel. But would be hard for a natural satellite to get into them.

I got this from the NASA page: Bizarre Lunar Orbits where they say

"There are actually a number of 'frozen orbits' where a spacecraft can stay in a low lunar orbit indefinitely. They occur at four inclinations: 27º, 50º, 76º, and 86º"

With moonlets of moons, the tidal effect isn't that important over short timescales. And after all even the Moon's orbit is not stable indefinitely, over billions of years.

SO CAN ANY MOONS HAVE MOONLETS?

Depending on the shape of the moon, and whether it has mascoms or not, a moon could in theory have stable moonlets for at least some period of time. But so far we haven’t found any moonlets of moons. There are some moons that could possibly have them in theory.

I answer this here: Robert Walker's answer to Do moons have moons?

Also wrote it up as science blog post: Can Moons Have Moonlets? Or Rings?

You can now get this as a kindle book along with many other answers on a similar vein, as part of

​Simple Questions in Astronomy - Surprising Answers

Or as a separate kindle book for just this one:

​Moonlets of Moons? Or Moons With Rings?: Can moons themselves have moonlets or rings?, Robert Walker - Amazon.com