Pre-World War II, this division between American civil and military society was less (perhaps, much less) of a problem. Then, the largest part of our (white male) population (of a certain age) was enrolled in our various state militias (or, their successors—the U.S. and state National Guards). Conscription, if not universal conscription, naturally flowed from actual congressionally declared wars. Likewise, a large swathe of Americans, across all social classes, could expect to see military service in war, including, unfortunately, Indian wars. (Lincoln and Davis both served in the Black Hawk War.)With all due respect, that's not right. After the Civil War, the United States military shrank to a very tiny size and remained that way until the First World War. The very brief Spanish American War (1898) aside, the main business of the Army was fighting unions, not fighting Indians -- the famous Indian wars were fought by Civil War veterans like Sheridan and Custer (who was one of Sherman's favorites). So it's not a new generation of fighting men, it's the same generation staying on in a much smaller army, fighting battles that were minor echoes of the great battles of their youth.
Military service was the exception rather than the rule for most of the 19th century, excepting the Civil War generation. It was again between the two World Wars. It is actually only after the two great wars -- the Civil War and WWII -- that we have had a very high percentage of veterans in American culture. You can see the effect of this by looking at this poll about the percentage of American men who are veterans. All the way to retirement age, the percentage hovers around twenty percent. It shoots up among the eldest among us, so that 80% of those 85-89 and 74% of those over ninety are veterans.
For women, too, we see the lingering effects of the draft because women were not drafted: overall, only 2% of women are veterans, compared with 24% of men, mostly in the older generations. That number is 14.5% currently, so the overarching importance of the WWII/Vietnam drafts is what is driving the percentage of female veterans down into the low single-digits.
Another way you can see the lingering effects of WWII and, to a lesser degree, Vietnam is in the chart on proportions of veterans by region. The numbers are fairly flat: overall 12.7%, with a low of 11.4% in the Middle Atlantic states and a high of 14.6% in the Southeast. That means we're talking about the bulk of these numbers coming from the draft eras: since the introduction of the All Volunteer Force, the military has been 40%+ from the South. The trend has only intensified since Heritage did that study in 2008: recent numbers show that it was 44% in 2013. People move around, of course, but the relatively flat percentage of veterans by region shows the lingering influence of the massive draft of WWII and the smaller but significant draft of Vietnam.
So really, if a culture is a way of life that one generation passes to the next, it doesn't make much sense to talk about an "American martial culture." There are some families, especially but not only in the South, where a culture of military service is passed from one generation to the next in the days of non-compulsory service. American culture overall has not been martial. For all but two generations, the bulk of Americans have not served in any military nor fought in any wars. The institutions have cultures, but American culture overall has been not much affected by them for most of our history.