Sam--
If indeed you meant Greenwich, CT, you might consider looking into visiting the charming Shennecossett Golf Course - the Donald Ross-designed (well, 15 of 18 holes, anyway) municipal course in Groton, CT, about 100 minutes east along the Connecticut coast. The 4th hole is a par three with one of the more severe greensites I have encountered on a long one-shotter, and there are a number of other solid golf holes on the property, plus some lovely views toward the end of the round.
Doing so would lift you out of the New York City environs and into a much more New England setting for the day, which you might find intriguing. The CT coastline receives quite little snow most winters, so there have been many winter days on which it's been one of the only CT courses open.
Should you like to make a day of it, you can also visit Fenwick Golf Course in Old Saybrook, which you would pass on your way to Shennecossett. It is regarded as one of America's most charming nine-hole courses. Likewise, it stays open on many winter days.
But, the possibility remains that the weather may shut you out of golf when you visit. But in that case, you'll have a number of other options. In NYC, the Metropolitan Museum of Art and Museum of Modern Art are two great places to spend a day, particularly if you have even the faintest interest in art.
Hope you enjoy your time stateside. Americans are actually not as brusque as they may sometimes appear on the Internet.