Possibly, because of what else the relay powers. It also supplies the power to the heating elements in the O2 sensors, and I think the supply voltage for the sensors themselves. Because they are so far downstream, they need the internal heaters in order to stay at operating temperature. If the heaters aren't getting power, they will generally not register properly and sense a lean-burn condition.... the result is that the ECU will send more fuel to the engine. Not to mention, they won't be able to self-clean anywhere near as good either.
I'd test the wire continuity between the relay and the o2 sensors first. When I get home, I'll dig out the wiring diagram and double check if there are any other switch-points to check on them.
My oil line fix
My 100k stopped odometer fix
Brace for impact, I'm in a "modifying mood". Massive thread count inbound.