A rescue mission, though risky, might have been possible, provided NASA management took action soon enough. It was determined that had NASA management acted in time, two possible contingency procedures were available: a rescue mission by shuttle Atlantis, and an emergency spacewalk to attempt repairs to the left wing thermal protection.
Video taken during lift-off of STS-107 was routinely reviewed two hours later and revealed nothing unusual. The following day, higher-resolution film that had been processed overnight revealed the foam debris striking the left wing, potentially damaging the thermal protection on the Space Shuttle. At the time, the exact location where the foam struck the wing could not be determined due to the low resolution of the tracking camera footage.
NASA's chief thermal protection system engineer was concerned about left wing TPS damage and asked NASA management whether an astronaut would visually inspect it. NASA managers never responded. The NASA Mission Management Team felt there was insufficient evidence to indicate that the strike was an unsafe situation, so they declared the debris strike a "turnaround" issue (not of highest importance) and denied the requests for the Department of Defense images.
Of course, now that NASA know they were wrong to ignore it, a great deal of the Shuttle's time in orbit is now spent checking all the tiles thoroughly.