Why do you want to invest in the European stock market specifically in the first place?
There's nothing that keeps non-US citizens from investing in US market ETFs.
That said, European economies are quite diverse. There are ETFs for the European market as a whole (Euro Stoxx 50, for instance) but they tend to fare worse than their national counterparts.
In addition to simply investing in US ETFs you might consider investing in European ETFs that reflect strong national markets. However, investing in European markets probably requires more research and market knowledge than the fire-and-forget approach US-based ETFs are famous for.
National NASDAQ equivalents such as TECDAX might be interesting, too. Keep in mind, though that these typically are much more volatile than NASDAQ because these indices don't comprise tech giants such as Amazon or Apple but much smaller companies which often are unknown to the general public but in many cases still are market leaders in their respective categories. Therefore the risk higher is higher - but so is the potential return.