Albeit with some erosion. To imagine the result of 500 years of lack of maintenance, take a picture of the current state of Mount Rushmore and apply a mild Gaussian blur filter.
We are talking about four gigantic faces carved in stone on the side of a mountain in a region that is not densely populated, filled with wildlife, far from industrial complexes (e.g. acid rains are unlikely), and surrounded by woodland, prairies, and badlands.
Interesting part, behind one of the faces there should be a room carved inside the mountain and filled with documents. It could probably be used for something else too.
Dams require constant attention. In 500 years there may be but a slight memory of it from some pieces of concrete attached to the sides of the mountain. The water will break through and erode its way across. A lucky dam will crack at the bottom and remain as an archway across a river. I doubt that is likely to happen, but if it does, here is a list of the tallest dams in the US.
A note about gravity dams. As suggested by Andon in the comments, gravity dams may be less prone to failures and thus longer lasting, perhaps even for the timescale of the question.
Skyscrapers are marvels of engineering, but they too require constant maintenance. Not much will survive 500 years due to the elements, corrosion of the internal supporting structure, and pillaging of construction materials. At best, the foundations of these tall buildings may remain as a sign of their greatness.
Bridges (Golden Gate), signs (Hollywood sign) or statues (e.g. Liberty Statue). These too require constant maintenance. Today the main threat is due to rusting and corrosion. In 500 years without a stable social structure, these constructions will be the number one source of free high-grade metals. Not much may be left, except perhaps the pedestals.
Other monuments and memorials
A quick look at the rest, taken from a subset of two wikipedia lists:
National Monuments, National Memorials
- High chance of surviving monuments, given that they already survived a long time on their own without proper maintenance (e.g. ancient precolombian dwellings); also, their being further away from bigger cities may spare them from pillaging or being used a quarries for construction materials; some could likely be buried under some meters of soil brought by rainwater, wind or partially damaged by minor land slides: Agua Fria, Aztec Ruins, Bandelier, Bear Ears, Chimney Rock, Fort Union(not really in working order), Hovenweep, Little Bighorn Battlefield Monument, Navajo, Poverty Point, Tuzigoot, Wupatki, Fort Caroline
- Probably still existing, at least in part, but being a good standing castle, or a defensive structure, possibly with thick stone walls stone, it might be used in between as a defensive structure and suffer significant damage; also, not negligible chances of the structure being completely demolished during some skirmish, especially near bigger cities: Castillo de San Marcos, Clinton Castle, Fort Frederica, Fort Matanzas, Fort McHenry, Fort Monroe, Fort Pulaski, Fort Stanwix, Fort Sumfer, Governors Island, Montezuma Castle, Salinas Pueblo, Chamizal, Lincoln Memorial, Jefferson Memorial,
- Probably still existing, but less likely to be accessible due to natural problems, such as earthquakes collapsing caves, or floods placing the grounds underwater; if no such events occur, then the monument should still be there as it has existed already for quite some time without maintenance: Glia Cliff Dwellings, Ocmulgee, Russell Cave
- Very unlikely despite being robust and sound structures, mostly due to location, accessibility as quarries, and not easily defendable unless major masonry work is undertaken: Lee Memorial, Federal Hall, Grant's Tomb, Korean War Memorial, Martin Luther King Memorial, Perry's Victory Memorial, Vietnam Veterans Memorial, Washington Monument, War World II Memorial