Was it deemed of as enough of a failure for not securing one or two of five beaches to call off the operation and extract the men?
Eisenhower's prepared speech says he would've recalled the invasion if he deemed they failed to "gain a satisfactory foothold". What was the cutoff point for a satisfactory foothold?
As you likely know, there were five planned beachheads. From east to west: Sword (British), Juno (Canadian), Gold (British), Omaha (US), and Utah (US). The British and Canadians met hard pockets of resistance, but managed to push the farthest inland on D+0. The American divisions at Utah also met relatively light resistance and suffered low casualties. The Omaha bridgehead, meanwhile, teetered on disaster. The Omaha sector included some of the most defensible terrain and earthworks (from the German perspective) and some of the most daunting geography and man-made barriers from the Allied perspective.
The Normandy assault was meticulously planned, and everything needed to go like clockwork. Portions of the armada and air support were designated for each beachhead. If Omaha failed, that would leave a significant part of the Allied armada stranded, so to speak. They could not easily turn and steam for Utah or Gold Beach because that would create a logistical nightmare in bad weather on rough water that could put thousands of lives in significant danger.
Abandoning Omaha would leave almost 40 miles between the British/Canadian zone and the lone American zone at Utah. Now we approach speculative territory. The German divisions concentrated at Omaha could have easily turned and counterattacked the Americans at Utah. They could have also attacked the British beachhead, but they likely would have turned on Utah because: (1) it was closer; (2) Utah was lightly defended and its garrison needed bolstering anyway; and (3) there was greater potential to overwhelm the Americans on one beach and close both Omaha and Utah down, thus leaving one large beachhead of British and Canadians to deal with.
The 1944 landings at Anzio were somewhat similar in that German counterattacks bogged down the initial success of the beach assault. Losing Omaha could have resulted in a similar scenario, albeit on a much larger and more terrifying scale. Even if the American beachheads failed and the Gold-Juno-Sword trifecta held firm, it would not really necessitate cancelling the entire operation. Eisenhower and his commanders already knew it would take the Allies several weeks to break out of the beachheads due to unforgiving Norman geography (see: hedgerows) and the concentration of German resistance. His prepared speech was likely written in the event that all beachheads failed, or that the casualties involved were deemed unacceptable. Part of the reason the Allies chose Normandy was because the Atlantic Wall was much stronger at other Channel crossings that were "easier" to attack, like Calais.
However, losing Omaha (and by extension maybe Utah) would have pushed the schedule back several weeks, resulted in higher casualties, and would have made D-Day nothing like the stunning success we know it as today.
EDIT: Another source - highly recommended.