When the U.S. forces invaded Okinawa, which is the smallest and least populated of the 5 islands that make up the country of Japan, we had 49,000 casualties of which 12,000 were killed.
Japanese soldier's rarely surrendered and fought to death. Plus the entire Japanese civilian population, every man, woman, and child, was being trained on the main islands to attack and kill American soldiers using sharpened sticks, kitchen knives, and metal garden tools.
The U.S. military commanders estimated we would have between a million to two million American soldier casualties, and many more millions of Japanese citizens killed when we invaded Japan.
So yes, the dropping of the two atomic bombs saved millions of lives. ...