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Some museums that are usually closed on Mondays are open on Monday holidays; the Met is one of them (they're open today, for instance). July 4 is a Tuesday this year of course; very few places will be closed.
Very quiet in NY that day except for the Macy's fireworks display which is free for the viewing on East Side of Manhattan. Get there early for the best standing spot.
Get there really early, bring water and snacks - since about a million people attend. And afterward it takes quite a while to empty out since the major roads are closed to traffic - and you have to walk.