First, assume it did. Higher oil prices have two effects: They push up prices, and unless you're an oil company, they hit corporate profits (check out the news from Ford and GM). Higher interest rates also push up prices, and, unless you're a bank, they also hit your profits. The result: stagflation -- inflation and rising unemployment at the same time. Raising interest rates after an oil shock was exactly the mistake the Fed made following the OPEC shocks of 1973 and 1979. In recent testimony before Congress, Bernanke tried to escape this trap, pointing out that price increases haven't yet passed from oil into the rest of the economy. So maybe he's for targeting core inflation -- price increases with the two "volatile" components of oil and food taken out. But here's the problem with that: Inflation has to start somewhere. Usually it shows up first in sectors where prices are, well, volatile. Only later does it move to sectors like government procurement, where prices change slowly. So if the target is shifted from actual inflation to core inflation, the Fed would be saying, "We'll fight inflation, but not until it's really entrenched." And if you wait until then, you've got a serious problem.
So it seems you're damned if you hurry and damned if you delay.
Bernanke, however, believes that targeting would help fight inflation another way: by sending a clear, transparent signal that would affect what economists call "inflation expectations." The idea is that workers and businesses don't just wait to react to higher interest rates. Rather, they set wages and prices partly with an eye to the Fed's policy stance, which they can read about in the newspapers or watch on TV. Thus, if they understand that the Fed plans to be a tough inflation fighter, then supposedly they will keep their wages and prices under control, so as to avoid the punishment of higher interest rates that they rightly fear.
This idea has a pedigree, and curiously enough, some of it is German. In arrangements that endured until about a dozen years ago, the Bundesbank negotiated in effect by threat and counterthreat with the largest trade union. If wage demands were too high, then the bankers would raise interest rates. The deutsche mark would climb, and the metalworkers, who produced much of Germany's exports, would lose competitiveness and their jobs. An atmosphere of credible threat, in other words, helped keep the workers in line.
But the German system worked (for as long as it did) because the union making the key wage decision also had the most to lose. Higher interest rates, as they affected the exchange rate and foreign trade, would hit them hardest and first. By moderating wage demands, they could keep Germany competitive and prosperous as a whole, and also keep their jobs. So they had something to gain by making a disciplined choice.
In the United States, we don't have a centralized bargaining process, with a strategic union faced off against the central bank. Here the signal would have to be transmitted by the central bank to the public mainly through the press, and implemented in very decentralized labor markets.