Not sure if it is that simple. If you take the relationship between the "real" dollar and oil going back to 1975 you have two noticeable spikes in the oil price and also two times where the dollar has traded near its highs for this time range.
The first time oil spiked was leading into a short recession in the US that started at the beginning of 1980. On this occasion we had a high oil price and a low US dollar.
Then the dollar spiked in June 1985 and it is clear it was not good for oil as it was trending down during this period.
That downward trend in the oil price continued during the nineties and the dollar also lost steam and both oil and the dollar traded low for the best part of that decade.
The US dollar was on the accent coming out of the nineties and peaked again in 2002 just after a short recession in the US which ended a year earlier. The price of oil once again was low while the US dollar was high.
The oil price spiked again during the GFC and the recession in the US that accompanied it. By historical measures since 1975 the "real" value of the US dollar was also low during that period.
We are currently at a point on the below chart where the oil price and the dollar are crossing once again. The last time this happened was in Feb 2005, but the cross was in the opposite direction which lead to the oil price spike in 2008. If the current cross continues it should lead to more dollar strength and oil price weakness.
However I think the US recovery is not that strong and a falling oil price combined with the dollar strength drag on emerging economies will weaken the US economy and will bring it into recession soon enough. If so the history of the chart below suggests that we could see another spike in the price of oil. Having said that however only two out of six recessions in the US since 1975 have been correlated with spikes in the price of oil.
So you can't necessarily rely on a US recession or the inverse correlation between the oil and the US dollar working every time. The price of oil is low now because of a US central bank engineered credit bubble that has lead to the misallocation of capital through the distortion of any sensible risk reward balance on the price of money. The ramifications of this are now being seen in the US shale patch as capital is starting to dry up rapidly. As a consequence of the falling oil price capital is also drying up for non-shale projects in oceanic basins where the costs are also higher. Eventually the oil supply and demand dynamics will restore the supply side of the equation back to a more historically familiar place where the US is no longer the swing producer and the Arabs will have more say over pricing. When this time comes IMO the Arabs will start to play fiscal catch-up by turning the spigot off once again.
Or maybe not.
Eshmun
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