Well, that's a *little* scary, but consider…
#1 the laser beam attenuates rapidly under water. Any satellite small enough to launch will be unable to damage anything underwater, only detect. If that.
#2 the laser can search exactly one spot at a time. This means it would take forever to scan an area, so it's a pretty terrible search device. If it works it's a great *tracking* device, which is very serious, but not as serious as a good searching device.
#3 only certain wavelengths can travel underwater well enough to accomplish this goal, and it wouldn't be tough to put sensors on top of the sub to detect those wavelengths. That means that, unlike sonar, the sub's crew will *know* when they're being tracked. Some clever person will figure a way to break contact and then they have to start searching all over again.
#4 if you paint the top of the subs the right color you should be able to greatly reduce the laser reflection. That would greatly reduce the depth to which this satellite could detect subs.
It feels like they're goal is a little optimistic. If we had time we could do some research and figure out how many watts of electricity a satellite can produce/consume, then figure out which wavelength they'd need to use. Figure out how efficient the lasers in that color are, then how much attenuation you'd get from the atmosphere and how much per meter of ocean. Then you'd know how sensitive the detectors would have to be and you could make a guess at whether or not this is practical.
But there are some *really* sensitive detectors around nowadays. You'd have a tough time getting the best ones working on a satellite, but I bet it's possible.
There are two more serious problems with that satellite, though.
First, the US will know where the satellite is. If there comes a day you need to blind it you just hit it with an inexpensive laser from anywhere in theatre. The direct beam from even a modest laser would overwhelm the detector and blind it to the extremely faint reflections from a submarine under the ocean. So when the satellite is most needed it would be useless.
Second, you need to know where your detector is pointing at the moment you get a reflection if you want to know where the submarine is. You would probably want these in a very high geosynchronous orbit so they could watch continually, but let's assume a very low orbit of 1000 miles. If the the satellite is angled off from what it's supposed to be just a hair, say 1mm for a 20' satellite, that similar triangles math we all learned shows us that the position of the sub is 250 meters off. That's not bad, but that's assuming a low orbit and a single millimeter misalignment. If it's a geosynchronous satellite and it's an inch off then your detection is 16 miles off. I guess that's still close enough to send in the helicopters to localize it.
But it's too easy to blind. That will be the Achilles heel.