The bill would enhance background checks for gun purchasers younger than 21, provide money to administer "red-flag" laws, and crack down on straw buyers.
A spate of mass shootings over the past few weeks have rocked the United States and fueled public frustration with longtime political inaction over gun violence.
A month ago, a gunman attacked an elementary school in Uvalde, Texas, killing 19 children and two teachers in a rampage that has triggered nationwide demonstrations against gun violence.
At least 21,000 people have died from gun violence across the United States so far this year, according to the latest data from Gun Violence Archive.
On Thursday, the U.S. Supreme Court struck down a New York state law that places restrictions on carrying a concealed handgun outside home.
The Supreme Court decision is "harmful and disturbing," Jack Resneck Jr., president of the American Medical Association, said in a statement.
"Firearm violence is a public health crisis, and easier access to weapons and fewer restrictions on who can carry them - and where they can be carried - are dangerous steps in the wrong direction. Overturning decades of reasonable firearm regulations will cost more lives," he said.