House of Representatives Speaker Paul Ryan said the executive orders, which bypass Congress, "undermined liberty" and would be challenged in court.
Republican presidential frontrunner Donald Trump said that, if elected, he would reverse the measures.
In an emotional address on Tuesday, Mr Obama accused the gun lobby of holding the country hostage.
Wiping away tears, he recalled the 2012 Sandy Hook primary school shooting in which 20 children and six adults were killed.
President Obama's executive actions involve:
He previously told the BBC that the failure to tackle gun control had been the greatest frustration of his presidency.
Why Obama is powerless - the roadblock at Congress
Are you mad or criminal? - the question a gun seller asks
Texas women and their firearms - a photographer taught to shoot at an early age
Do tighter gun laws work? - a state where guns are a way of life
Guns at home - the question parents hate to ask before a playdate
However, the largest gun lobby group, the National Rifle Association (NRA), said Mr Obama's steps would not have prevented any recent mass killings in the US.
"Rather than focus on criminals and terrorists, he goes after the most law-abiding of citizens," said Mr Ryan, a Republican.
"His words and actions amount to a form of intimidation that undermines liberty."
Correspondents say the announcement is already shaping up to be an issue in the 2016 presidential election.
Donald Trump said that he would "un-sign" the measures if elected to the White House - a sentiment echoed by other Republican presidential candidates.
Senator Ted Cruz tweeted that the executive actions were unconstitutional, with a link to sign up for his campaign correspondence on a web page that says "Obama wants your guns".
Former Florida governor Jeb Bush, also a Republican candidate, tweeted that he would repeal the actions and protect the Second Amendment.
Leading Democratic candidate Hillary Clinton tweeted: "@POTUS is right: We can protect the second Amendment while protecting our families and communities from gun violence. And we have to."
President Obama announced the law change at the White House, surrounded by survivors and relatives of victims of shootings.
"The gun lobby may be holding Congress hostage right now, but they can't hold America hostage," he said.
Gun violence is significantly higher in the US than in other advanced countries, killing about 30,000 people each year.
Sales of guns in the US appear to have risen recently amid speculation that the White House was going to tighten the law.
Shares in gunmaker Smith & Wesson rose to their highest value since 1999 ahead of President Obama's announcement.
Congress has been reluctant to pass any laws restricting gun ownership, facing pressure from gun owners and the NRA.

Images of a toddler, washed up on a beach in Turkey, ram home the sometimes deadly consequences of what is going on.
Right now though there seems little prospect of the UK government shifting its position on taking in more refugees, as EU leaders and Labour opponents have been calling for.
Although these latest images are heart wrenching, David Cameron's conviction is that it's more effective to tackle the cause not the symptom of this crisis.
That means substantial amounts of taxpayers' money going to provide food and shelter to refugees on the edges of Syria, to help on the ground. And diplomatically pushing for a political process that can move that country towards peace.
It does not mean throwing open the doors to many more Syrian refugees.
But there are two other parts to the logic of sticking to the position. First, despite what's going on, there is not huge public appetite for the government suddenly to act.
Second, there is a caution about sending a signal to other kinds of migrants, rather than refugees seeking safety, that makes the UK seem a more attractive destination at a time when the government is trying to limit the numbers.
Could that change? Of course. The intensity of the need means the calls to "do more" grow louder every day.
The public imagination, and that of Conservatives, could be stirred by the images to the extent that a shift becomes inevitable. As politicians drift back after their summer break the volume of voices might become too intense.
Or, the incentive to take more refugees as part of a wider deal with Europe - quid pro quo - might become tempting too.
Yet at this moment, there is nothing to force the prime minister to budge. But don't be surprised if the pressure on him to shift continues to build.
Leading US Republicans have denounced President Barack Obama's move to tighten gun controls.
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Only the hardest of hearts would not be moved by the latest images of the people who have been caught up in the danger and chaos of this mass movement.