House Budget Committee Chairman Paul Ryan (R-Wis.) said Sunday that he thinks deep, automatic, across-the-board spending cuts will happen because of Democratic opposition to GOP alternatives.

"I think the sequester is going to happen," Ryan said on NBC News's "Meet The Press."

The sequester -- which is the roughly $1.2 trillion worth of automatic spending cuts set to soon hit the Pentagon and other government agencies if lawmakers don't act -- is likely to happen, Ryan said, because of opposition from Democrats to Republican proposals for replacement cuts.

"We think these sequesters will happen because the Democrats have opposed our efforts to replace those cuts with others and they've offered no alternatives," Ryan said.

Ryan also echoed other GOP leaders as he took a stern posture against Democratic calls to discuss the need for new revenue in the nation's budget, arguing that the matter was addressed in the recent deal to avert the "fiscal cliff," which raised tax rates for the wealthiest Americans.

"Well, we already offered that back in the fiscal cliff negotiations. The point is, though, the president got his additional revenues. So that's behind us," Ryan said.