Why was the US unable to stay out of European wars during the adminstrations of President Washington through President Madison?
The basic reason for this is that the United States needed to trade with European countries. Without European trade, the American economy would have been badly damaged. This trade led the US to get entangled in European wars.
After the French Revolution, conflict arose between Great Britain and France (and other European nations). This conflict caused major problems for Americans. The US wanted to (and needed to) trade with Europe. However, both Britain and France blockaded one another and claimed the right to seize neutral ships trading with their enemy. This, of course, affected the US.
So the US was caught up in European wars. It could not stop trading (the effects of Jefferson's Embargo Act showed that), but its trade was constantly disrupted by French and British seizures of US ships. The US saw this as a violation of its neutrality rights.
Caught in this quandry, the US had a number of conflicts with France and Britain, finally culminating in the War of 1812.