The euro/dollar exchange rate has fluctuated between 1.50 and 1,20 over the last eight years as the market was trapped in this wide range without trending in any major direction . The dollars trending strength commenced in May of 2014 and accelerated breaking the 1,20 psychological and technical support as the market priced the comparative growth differentials between the Euro zone and the US. The Feds announcement of the possibility of a slight tightening of future conditions was a policy response to the strengthening economy and thus reinforced the dollar's strength (without putting a damper on growth), whilst the ECB's QE program, a reluctant response to the depressed state of growth in the Euro zone, added further momentum to its direction. Without doubt therefore the dollar's strength was mostly fueled by the perception of comparative growth differentials between the two continents. Clearly, this was enhanced by the existential problems of the Euro and Greece's predicament.
Now will this large exchange rate move put a damper on US's growth? Well, to the extent that the stronger dollar will lower the competitiveness of US exports and increase that of imports thus worsening the US's trade balance and hence its GDP, it will. The US being an open economy is thus likely to be adversely affected by its strengthening exchange rate.
The counterargument here is that QE should bolster Euro zone growth & will thus help the US's trade balance as Europe increases consumption and thus imports too. However, this will need to play out especially to see whether ECB QE does indeed increase growth. However, for as long as Germany and the rest of the core Euro economies continue to have negative yields on their debt instruments, growth prospects will remain bleak and hence the strong dollar will withdraw growth from the US economy.