Texas Sen. Ted Cruz was perhaps Obama's harshest critic, accusing the President of attempting to take away Americans' guns.
On Twitter, Cruz called Obama's actions -- which include expanding background checks on gun purchases and increasing funding for mental health and law enforcement agencies that perform those background checks -- "unconstitutional."
In an image included in the tweet, Obama was portrayed wearing a crown, alongside the message: "Gun control is government control."
It linked to a Cruz campaign website featuring Obama dressed in military-style garb, next to the headline, "Obama wants your guns."
Donald Trump's campaign manager, Corey Lewandowski, told CNN's Jake Tapper on "The Lead" that Obama is "making an end-run around the Constitution, trying to go directly to executive order and restrict one of the basic, fundamental principles of our country."
And Trump himself responded during a rally Tuesday evening, "Look the Second Amendment is so important. It's so important."
"They're not going to take your guns away, folks. They're trying," he said.
Florida Sen. Marco Rubio tweeted a link to an appearance on Fox News in which he characterized Obama's moves as "just one more way to make it harder for law-abiding people to buy weapons or to be able to protect their families. It's going to do nothing to prevent violence or crimes."
Former Florida Gov. Jeb Bush, meanwhile, tweeted a link to his op-ed in an Iowa newspaper in which he said Obama's actions trample on the Second Amendment.
Retired neurosurgeon Ben Carson tweeted: "The President's actions have everything to do with advancing his political agenda & little to do with actually protecting American citizens."
And former Arkansas Gov. Mike Huckabee used Obama's speech to push for restrictions on abortion -- saying that Obama's case for saving lives should apply to that issue as well.
Democratic presidential front-runner Hillary Clinton, meanwhile, vowed in New Hampshire on Tuesday that she will "take on that fight" and continue Obama's gun control push if she's elected.
On Twitter, in a tweet signed "-H" to indicate it was written by Clinton, rather than her staff, the former secretary of state thanked Obama "for taking a crucial step forward on gun violence. Our next President has to build on that progress—not rip it away."
And her campaign highlighted Republican candidates' criticism of Obama's comments on its website, warning that a GOP president would undo Obama's actions.
Vermont Sen. Bernie Sanders said in a statement that he, too, would keep Obama's actions in place if he's elected.
"It's become clear that no mass shooting, no matter how big or bloody, will inspire Republicans to put children and innocent Americans over the interests of the NRA. They are simply more loyal to gun lobbyists than our children. That's why I support President Obama's executive actions to make our communities safer," Sanders said.
And former Maryland Gov. Martin O'Malley vowed to build on Obama's actions.
"As President, I will build on President Obama's progress by taking further executive actions to reform our gun laws. These include using the purchasing power of our federal government -- the biggest customer gun companies have -- to advance gun safety, banning 'cop killer' ammunition, declaring blanket immunity for gun dealers and manufacturers unconstitutional, and ceasing to defend the federal immunity law," he said in a statement.