Pretty bad. Fort Vaux had fallen, Fort Souville was the scene of fierce fighting, and Petain's requests for a relief offensive became more frantic. It was in late June that Robert Nivelle gave his famous exhortation, "Comrades, On Ne Passez Pas!" (They Shall Not Pass!).
Had the Germans been able to make another push, the heights over looking Verdun would have been well within reach. The French positions below, along with the city, could have been brought under heavy artillery fire, and La Voie Sacre would have been under great pressure. Faced with such a disadvantageous position, a French retreat to the Meuse likely would have been in the cards.
It's also worth noting that Falkenhayn was fully expecting an attack on the Somme. It was the only place the British and French could launch a cooperative attack, there were reports in the Allied press, and the massive troop build up around Albert was hard to conceal.
Unfortunately, he was NOT expecting the offensive that followed. The expectation was an attack by one British Army north of the Albert-Bapaume road, and the plan was that c. 20 division of the German Sixth Army would counter-attack between Arras and the Somme, throwing the British off balance and inflicting great losses. Similar, much smaller attacks had been made against British forces at Mont Sorrel in the Ypres area, and at Vimy Ridge. The former battle had actually seen Major-General Malcolm Smith Mercer, GOC 3rd Canadian Division killed in action.
What Falkenhayn and the Germans got was an attack on a thirty kilometer front, straddling the Albert-Bapaume road and the Somme river, with XIII, XV, X, III and VIII British corps attacking north of the Somme, and a French army, the 6th, attacking south of and astride the Somme river. XIII and XV corps took the German first positions south of the road, and opposite the French 6th Army, the ENTIRE German first position was lost. South of the Ancre River, British forces had VERY mixed success, but inflicted losses on the enemy. July 1st, 1916, is remembered by the British as a bloody day of failure. For the French, it was a day of stunning, hard won and gratifying success. For the Germans, it was the day they lost the initiative.
On July 11th, 1916, operations at Verdun were shut down indefinitely, and the extra divisions of the German 6th Army to the north were siphoned-off, into 'das blutbad' on the Somme.