Did the Germans use bombs on the Americans on Normandy beach?

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Yes they did, albeit keep in mind the Battle for Normandy started on June 6, 1944 and lasted a good week... it then developed into the Battle for Brittany.
The following from War Diaries gained from various Luftwaffe units:

"The FW 190s of III./S.G. 4 were active against the landings on D-Day, flying three missions in the late afternoon. III./S.G. 4 claimed two direct hits on 400-ton landing craft off St. Aubin at around 17:40."

"we were woken at 04h30 and taken to the airfield from the hotels in the town (Nancy) where we were quartered. We were airborne a short while later and flew to Creil (north of Paris) at around 05h00 to have our Fw 190s fitted with underwing rocket launchers. We took off again at 09h30 to strafe shipping off 'Gold' beach. There was 7/10 cloud cover as we overflew the Seine estuary, which allowed us to close on our targets and launch our rockets. We could see a huge number of enemy fighters orbiting over the landing beaches. My rockets probably scored a direct hit on a "Victory" class troop landing vessel...we fled the scene and returned to Chamant near Senlis (south of Creil )after this sortie.."

Source:
http://www.feldgrau.net/forum/viewtopic.php?f=40&t=26632

"Actual results, however, by no means tallied with these reports, being considerably lower. Contemporary British sources (War at Sea Vol. V) mention the following sinkings by the Luftflotte:
On 6 June the frigate Lawford
On 10 June the American merchant ship Charles Morgan 7244 tons
On 13 June the destroyer Boadicea by air torpedo.
Damaged: On 7 Juno HMS Bulolo and the cruiser Eucrates. Losses of small craft are not reported

Source:
http://www.history.navy.mil/library/online/gaf_invasionnormandy.htm

The types of bombs and rockets used varied. The ME-109 and FW-190 carried two types of bombs namely one 1,100lb bomb and two 550 lb bombs.
http://www.historyofwar.org/Pictures/fw190_montecorvino.jpg

They also could carry rockets in under-wing pods:
http://i53.photobucket.com/albums/g64/PoorOldSpike/sub4/FW190-rocket.jpg...Show more

No. The Allies enjoyed such total air superiority that the only appearance by the Luftwaffe was a brief strafing run by two FW190s - and even these only hit the British/Canadian beaches....Show more