Clinton told "Fox News Sunday" host Chris Wallace that Trump's remarks -- which the Republican nominee has said were sarcasm -- are a national security problem.
Russia's suspected role in the hack of the Democratic National Committee's emails "raises serious issues about Russian interference in our elections, in our democracy," Clinton said in her first interview with Fox of the 2016 campaign.
"We would not tolerate that from any other country, particularly one with whom we have adversarial positions," she said. "And for Trump to both encourage that and to praise (Russian President Vladimir) Putin, despite what appears to be a deliberate effort to try to affect the election, I think raises national security issues."
In the interview, Clinton used the email issue to cast Trump as overly cozy with Putin -- part of an effort to undercut Trump's qualifications on foreign policy and to suggest he's not steady enough for the presidency.
"I think if you take his encouragement that the Russians hack into American email accounts, if you take his quite excessive praise for Putin, his absolute allegiance to a lot of Russian wish list foreign policy positions, his effort then to try to distance himself from that backlash -- which rightly came not just from Democrats, but Republicans, independents, national security and intelligence experts -- leads us, once again, to conclude he is not temperamentally fit to be president and commander in chief," she said.
Clinton also responded Sunday to Pat Smith, the mother of Sean Smith, who was killed in the attacks on the US diplomatic compound in Benghazi, Libya, and who said at the Republican National Convention that Clinton "lied to me" and that she blames Clinton personally for her son's death.
"My heart goes out to both of them. Losing a child under any circumstance, especially in this case -- two State Department employees, extraordinary men, both of them, two CIA contractors gave their lives protecting our country, our values -- I understand their grief and the incredible sense of loss that can motivate that," Clinton said. "As other members of families have lost loved ones, that's not what they heard. I don't hold any ill feeling for someone who in that moment may not fully recall everything that was or wasn't said."
Pressed on gun rights, Clinton said she believes the Second Amendment does guarantee individuals' rights to own firearms.
"Yes, but that right, like every other of our rights, our First Amendment rights, every right that we have is open to and even subject to reasonable regulation," she said.
And she discussed accepting the Democratic nomination Thursday night in Philadelphia -- saying she was emotional when she stepped onto stage after being introduced by her daughter, Chelsea Clinton.
"It was way over-the-top emotional," Clinton said. "My biggest concern going out there to make that speech Thursday night was whether or not I could control my emotions."
Her concern, she admitted, was "that I would start crying ... Watching my daughter and having her waiting on stage when I came out, I was pretty concerned about whether I'd make it through the speech."