Most oil is traded in US$. So, with higher oil prices, people will buy more dollars (increasing demand) to pay for fuels, thus stabilizing the $?
I guess you could say in nominal terms the dollar has more $$ demanded but its value isn't going up. Mostly only the US has to pay higher gas prices, the other currencies have all appreciated against the dollar. So in the end they use the same amount of their money to buy more shiny american coins to pay for the same amount of oil.