I remember hearing that the majority of American soldiers on D-Day had never before seen combat.
It's true that green troops were the majority, but there was a sizable minority of veteran units and troops in the early waves at D-day.
Here is a simplified order of battle for the first and second waves in the American side of D-day, not including airborne troops, or miscellaneous DD Tank Battalions and Combat Engineers.
Utah Beach:
8th Infantry Regimental Combat Team of 4th Infantry division (Green)
Omaha Beach:
116th Infantry Regimental Combat Team of 29th division (Green)
16 Infantry Regimental Combat Team of the 1st division (Veteran)
2 Ranger Battalions (Green)
So the only veteran regiment in the first/second wave is the 16th Infantry Regiment of the 1st Infantry Division, which has roughly the pedigree that Tom Hanks character, Captain Miller, has in the movie, being veterans of the North African and Sicily invasions. If I recall the movie correctly I believe Captain Miller is supposed to be from a Ranger battalion, and I don't think the Ranger battalions at D-Day had been in the Mediterranean or seen combat before June 6th.
In any case, it's probably fair to say that the main assault units at D-Day were about 1/3 veterans of the Med and 2/3 green units who had intensively trained amphibious operations in Britain for approximately 1 year. However, when we get down to the level of individual people, the veterans start to get even thinner on the ground. Most of the attached specialized units like DD tanks, Combat Engineers, Beachmasters, and similar were unblooded. Furthermore although the 1st Infantry division "Big Red One" is a veteran unit with veteran leadership, it doesn't follow that all of it's troops were all veterans of North Africa and Sicily, a portion of them would have been replacements of casualties or men promoted into other units.
All in all, it's difficult to say what proportion of men were veterans, but I'd say well under 30% of men who hit the beach before noon on June 6th had served in the Med.
Still it's enough of a proportion to give cover to a fictional Hollywood portrayal.
Two of the D-Day US divisions had already fought in WWII. The 82nd Airborne Division had made combat drops in the invasions of Sicily and mainland Italy in 1943. The 82nd (and the green 101st Airborne Division) attacked on D-Day via parachute and glider; they weren't part of the beach landings.
The 1st Infantry Division had landed in Algeria during Operation Torch, in 1942. They fought in North Africa into May 1943. Next, they took part in the invasion of Sicily, making a strongly opposed landing at Gela. They fought in Sicily to the end of the Sicilian campaign, and were then withdrawn to England to prepare for the Normandy landings. The first wave at Omaha Beach consisted of two regimental combat teams, the 116th RCT of the green 29th Infantry Division and the 16th RCT of the veteran 1st Infantry Division. The 16th RCT included many veterans.
Also in the first wave were tanks from the green 743rd Tank Battalion and three companies of Rangers. In Saving Private Ryan, Captain Miller was the commander of one of these Ranger companies. The three Ranger companies in the first wave at Omaha Beach were from the 2nd Ranger Battalion. The 2nd Ranger Battalion was new, and hadn't seen action. Some of the personnel may have come from the earlier Ranger battalions (1st, 3rd and 4th) which had been disbanded after two of them were destroyed at Anzio; most of the 400 remaining Rangers went into the 504th Parachute Infantry Regiment, but some might have gone into the new Ranger battalions which were formed.
I don't know if either of the real-life soldiers who inspired the character of Captain Miller (Captain Ralph E. Goranson and 1st Lieutenant Sidney A. Salomon of C Company) had seen action before Normandy.
For the organisation of the Ranger units in the 1st wave at Omaha, see: