No, you can only trade in the currency of the exchange the security is listed on. If it is listed on a US exchange, then it is priced and traded in US dollars.
If it is a European company, for example, a German company, it would have listed stock trading on European stock exchanges and would be priced, and traded in Euros.
The company might also list shares in the US, which are called ADRs (American Depository Receipts). Those shares are priced and traded in US dollars. The price of the ADR is linked to the price of the Euro-denominated shares traded on the German stock exchange. ADRs are an example of a depository receipt. Be aware that not every non-US company chooses to offer an ADR! If they do, you can find them on the NYSE or possibly the NASDAQ.
There is another special case: cross-listed shares. Here too, the currency of the exchanges is the currency in which the stock is traded.