I don't see why not. Hell Gate is fine if you go with the tide. (Navigating the East River - Planning Your Passage) You'll want an outboard in case the wind dies and the tide is going for both the East River and the Hudson. The East River is generally no fun, so stick to the Hudson or the LI Sound region.
I learned to sail on the Hudson, you should not just learn by the seat of your pants there. Thee is a lot of commercial traffic that cannot stop or maneuver around for you, you don't want to get stalled with your sheets all tangled in front of a barge.
I wouldn't take a 15' boat through the Hell Gate, or in the East River at all, on any regular basis. The problem is not just the currents, it is the traffic in a confined waterway, and the waves themselves. You can get 2-4' tall square standing waves reflecting up from the bottom in the East River, and that's a rough ride for a 15' boat of any kind.
If you know what you are doing, and you are very much aware of the conditions, it certainly could be done safely. But if you have to ask, no, stick to the Sound or the harbor and skip the East River entirely for now.
And even in the harbor--remember that traffic will have a hard time seeing you. If you can't get out of their way, there may be nothing they can do to avoid running you down. There's much more room in the Sound, and bays where you can have plenty of fun without the current or traffic.
It would not be choice of vessel. There can be an incredible amount of traffic through NY harbour including (but not limited to) ferries, excursion boats, tugs, barges, cruise ships). Many make no effort to get out of your way when you are sailing. We sailed there in a 45 footer with AIS, good speed under sail, and a good engine. Also the winds tend to be a bit flukey and there are significant tides. I think you should say away from the harbour in that boat or get a different boat if you want to be in the harbour.
i'd say the WW15 is a great beginner's boat, but you need to stay away from demanding conditions until you have some experience. Conversely, a 45-footer might be ideal for Hell's Gate but it's a poor choice if you're just starting learning to sail.
I'd also strongly suggest some ASA or similar lessons, perhaps 101, then get a little experience sailing the WW15, then do 103 - for me this meant getting the best out of the 103.
The "NYC area" is a pretty vague term. It sort of goes from perhaps Nyack, on the Hudson, down to Sandy Hook, and from Brooklyn through Hell Gate up and out to Long Island Sound. The WW 15 is a fine boat, but probably isn't well suited to the conditions you'd find in most of that territory most of the summer sailing season. Typical southwest breezes that might be nice and strong in Sandy Hook or even in NY harbor itself get diverted by the city's thermal updraft and don't come down again until they get to Connecticut - often the other side of New Haven. The Hudson might fare somewhat better, but generally wind is light in much of this area. The WW15 has a relatively small sail area for such conditions. It will sail, but might not perform as well as another boat with bigger sails. Powerboat wakes can make sailing an undercanvassed boat very frustrating. If that's the boat you have and you need trailerability, you can make it happen. There might be other boats better suited to the conditions you're likely to come across, however.
Where in the NYC area are you? As mentioned, traffic is a bigger challange that currents. A 15' boat with a small outboard may not be the best thing to dodge super tankers in. That being said, there are lots of great spots to sail small boats in in the NYC area, Sandy hook bay and the Navesink being two. Drop me a PM.
Getting in and out of the Bayside Marina, accessing the map there and figuring out how to hike out to where you can park the car, can be a real RFPITA but you may find the water better than at the airport.
Also bear in mind that if you sail in the harbor and cross that faint blue line into NJ waters, you DO need a formal operator's certificate to operate any type of motor(ized) boat in NJ waters.
Long Island Sound has lots of interesting places to visit within range of a weekend cruise for you. You will want to work with the current to get in and out of the World's Fair Marina, or you will spend a lot of time not getting much of anywhere. Buy a copy of Eldridge's Tide & Current book and make sure your motor is in good shape with plenty of fuel. Fair winds!
I would think that there are better options than worlds fair to put in, if you are trailering that will allow you to sail sooner without having to motor out and worry about strong currents. Something on Manhasset bay Perhaps?
Haven't been down there in a while, and I have no idea if you can get a mooring except through the various clubs. But where else can you get to a mooring by public transportation and have such a wide range of restaurants to visit before heading home. (G)
There's also the Gateway Marina, across the road from Floyd Bennett Field, lots of fairly shallow and protected water, and that's also a short bus ride form the subway or an easy car trip, although it is more in the middle of nowhere for any other purpose.