The euro rose 0.37% to $1.0528 as the single currency responded with relief that Trump did not mention European trade.
Shares of European car manufacturers fell sharply on Tuesday as they have huge production capacity in Mexico that is sold to the United States.
Against its Canadian counterpart, the dollar remained almost unchanged at 1.4054 Canadian dollars, after touching a 4-and-a-half-year high of $1.4178 on Tuesday.
The dollar has remained well off its highest level against the Mexican peso since July 2022, and was last down 0.20% on the day.
The New Zealand dollar rose after the Reserve Bank of New Zealand cut benchmark interest rates by 50 basis points to 4.25% while noting that inflation fell to nearly the midpoint of its target range.
The Australian dollar reached $0.6480, up 0.10%, after domestic consumer price inflation remained at its lowest level in three years in October.
In cryptocurrencies, the price of Bitcoin rose 1.8% to $93,271, which is well below its recent highs.
