Speaking in Cheyenne, Cheney said it's time for Republicans in Congress to stop "cutting deals" with Democrats.
Cheney did not once mention Enzi by name or indirectly during the 15-minute news conference. Instead, she focused a series of blistering attacks on President Barack Obama, accusing him of working to disarm the country and nationalize much of the economy.
"We've got to stand and fight, and we have to defend what we believe in. We have to not be afraid of being called obstructionists," Cheney said.
"In my view, obstructing President Obama's policies and his agenda isn't actually obstruction; it's patriotism," Cheney said. "I think we have to stop what he's doing, and then as conservatives, we've got to say, 'Here's what we believe,' and, 'Here's the path forward,' and that's what I intend to lay out in this campaign."
Cheney didn't give specifics in response to reporters' questions about what she believes Enzi might have done to show complicity with Obama's agenda.
But she said, "Instead of cutting deals with the president's allies in Congress, we can be opposing them every step of the way."
Cheney would not answer if her "cutting deals" remark referred to Enzi, who has expressed pride in being able to work with Democrats on some issues. He declined to comment about Cheney's run against him.
Cheney's father is still widely admired in his home state, yet some Republicans already are lining up behind Enzi. They include Wyoming U.S. Sen. John Barrasso and the state's lone U.S. representative, Cynthia Lummis, as well as the National Republican Senatorial Committee led by Sen. Jerry Moran, R-Kan.
Cheney downplayed her opposition from fellow Republicans.
"I am really going to work hard for the votes of the people of Wyoming. I think that the people of Wyoming are pretty smart about these things," she said by phone after the news conference.
"When you hear things like the NRSC is behind Sen. Enzi, I think that frankly a lot of the response of the people of Wyoming is: 'We're going to decide this. Nobody from outside the state is going to tell us how to vote.'"
Asked about gay marriage -- her sister Mary is a lesbian and Dick Cheney has openly supported gay marriage -- Liz Cheney said the issue should be left to voters in each state to decide by referendum.
A Republican primary brawl could help a Democrat seeking the office, but only if the Wyoming Democratic Party is able to transform itself quickly from the weakling it's become in recent years.
No Democrat has expressed interest in running for Enzi's seat next year, and no Democratic candidate has been competitive in a statewide race since two-term Democratic Gov. Dave Freudenthal left office in 2010.
Wyoming Democratic Party Chairman Pete Gosar, of Laramie, said he sees little difference between Enzi and Cheney. Gosar ran unsuccessfully for his party's nomination in the 2010 Wyoming governor's race and works as a pilot for the state.
"Until I see some sort of people-centered, issue-based solution agenda, then I don't know that somebody who ascribes 100 percent with the national Republican philosophy is any different from someone who ascribes 99.9 percent with it," Gosar said. "So they look very similar to me."
Robin Van Ausdall, executive director of the state Democratic Party, said the party is looking for viable Senate candidates. She declined to name potential contenders.
She called Cheney "an outsider, and very extreme."
"I do think that it does present an opportunity for Democrats," she said.
Copyright 2013 The Associated Press.