Is one allowed to go out watching the fireworks in the new year of the common era, is there any reason, why it could be forbidden?!
See this article which analyzes part of Igrot Moshe on the topic of New Year's celebrations in general.
While you should read the whole article, I'm excerpting the part that I believe is most relevant to your question:
Rabbi Feinstein (Iggerot Moshe, Yoreh Deah 4:11(4)) is logically correct in his observation that:
‘Thus, it is obvious in my opinion, that even in a case where something would be considered a prohibited Gentile custom, if many people do it for reasons unrelated to their religion or law, but rather because it is pleasurable to them, there is no prohibition of imitating Gentile custom. So too, it is obvious that if Gentiles were to make a religious law to eat a particular item that is good to eat, halacha would not prohibit eating that item. So too, any item of pleasure in the world cannot be prohibited merely because Gentiles do so out of religious observance.’ ”
My opinion, which I feel is supported by the above quote - you are watching fireworks, which per se is not any form of religious observance or practice. You could technically claim that that you are doing so completely unrelated to the New Year's celebration, i.e., it's just fireworks - no different from your watching Independence Day fireworks (i.e. - your disconnecting your watching from the day that it occurs.) There is certainly no problem doing this.
In the worst case, if even in your mind you are celebrating New Years, see the rest of the article that cites most opinions stating that New Years has lost its religious significance.
FYI - There will be a fireworks display just before midnight tonight (1/1/15) over New York Harbor at Liberty Island.