I don’t think that there will be a war.
However, the USA represents now half of the world market capitalisation, hence whenever the USA stock market goes down European stocks also go down. Another reason is that it is a very big market and many European companies have interests there, whether it is financial companies, utilities, industrials, etc. In the same way lots of USA companies have large interests in the EU, including Tech such as Apple/Microsoft/Amazon/Netflix, consumer products such as Coca-Cola or Kellogs, industrials such as Ford or financials such as investment banks. The world is very much connected, particularly amongst the most developed economies.
If for example the USA has very large problems, due to Covid-19 or otherwise, and Europe does not, the European stock market would probably do significantly better, but it would be likely that neither of the 2 would do very well.
Regarding your comment on the US Dollar, I would like to see European currencies, in particular the Euro becoming the “go to” currency worldwide in the near future, however it currently does not seem very likely, at least not in the short term. For example, with most debt worldwide issues in Euros, most ETFs listed in Euros and the Euro being the benchmark for most currency exchanges worldwide… it may happen one day, but not soon. I think that for that to happen the EU has to become more united and move towards being more of a federation and less independent states.