At least 40 people were on board when the bus went off the road and crashed into a ravine on Tuesday night in the Pauri Garhwal area, according to officials.
Following an overnight operation run by the state's crisis response force, 21 passengers have so far been saved.
In a tweet, the prime minister expressed his "thoughts are with the grieving family."
When the tragedy occurred, the bus was transporting a wedding procession from Laldhang to the village of Bironkhal in Uttarakhand.
Although the cause of the accident was not immediately clear, state authorities stated they were looking into it.
On Wednesday, senior police official Swatantra Kumar Singh informed the ANI news agency that a rescue effort was still in progress.
Pictures taken at the scene showed the bus's wrecked remains laying next to a steep hill while rescue workers assisted in removing the survivors.
Pushkar Singh Dhami, the chief minister of Uttarkhand, stated that the rescue teams were receiving all assistance that could be given to support their work.
India is infamous for its terrible road safety, with over 100,000 people dying annually in collisions brought on by reckless driving and crumbling infrastructure.
In India, particularly in the smaller towns, buses are a well-liked form of transportation. However, owners frequently break safety laws and over-pack vehicles.
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