Yes and No.
Non-military obstacle courses do exist, but most are not "open" to the public. They are privately owned courses that require you to pay to use them. Most of these are what I would consider, challenge courses and are not very long, but have a large number of height challenges that require teamwork and coordination. These facilities are used as team-building activities by groups and companies and are typically rented for a specific time period.
What you're looking for, a course that is several miles long with intermittent obstacles does not exist on a permanent basis. At least none that I have found in most areas. Which area of the US are you looking at? Some areas may have trails/runs which could almost classify as a military style obstacle course, but this would vary greatly depending on location. (For instance, here in Washington State, some parks are set up with obstacle courses for public use, but the obstacles are not very advanced. As noted in a different comment, this is due to liability.)
However, these types of obstacle courses do exist on a less-permanent basis for certain events. Spartan Race, Warrior Dash, Mud Runs, ect... all exist in most areas and will be a military style obstacle course which, after paying, are open to the public. However, these courses only exist for a limited time and won't be there for "training" purposes. I would suggest looking into these types of events, if a military style run is what you're after. But I have not heard of these types of obstacle courses being setup permanently on a trail for any length of time. Beyond the aforementioned liability concerns, these types of courses destroy the trails they are on.