Rep. Steny Hoyer, D-Md., the No. 2 Democrat in the House, said Tuesday he expects a new battle over voting rights to play out similarly to the current fight over immigration, saying it was "in the best interest" of Republicans to act on both or risk losing even more minority support.

Hoyer spoke shortly after the Supreme Court announced Tuesday that it was striking down a section of the Voting Rights Act that requires mostly southern states to get Justice Department approval before making even minor changes in their election process. In a 5-4 ruling, the court said Congress could rewrite the formula that determines which states must get federal clearance, but negated the portion of the law that targeted nine states that in 1965 were judged to be obstructing minority voting.

Divisions within the Republican Party and the glacial pace of Congress make it difficult for any legislative action that would single out which states still discriminate against certain voters. But Hoyer said that just as in the immigration battle, there will be mounting pressure on Republicans to act or risk appearing insensitive to minority issues.

"There continues to be subtle ... ways and means to preclude or discourage or disenfranchise people that people think are going to vote against you. That's wrong," Hoyer said. "Ultimately, it will be in the best interest of their party" to tackle these issues.

The Voting Rights Act was reaffirmed by a Republican-controlled Congress in 2006 with strong bipartisan support. The makeup of the GOP is "much more ideological" now, Hoyer said, but Republicans with national ambitions are helping to engage the party on issues like immigration.

"There is real pressure in the Republican Party on the presidential wing side that is thinking nationally," Hoyer said.

He added that he was hopeful immigration reform legislation could pass in the House in July. The Senate is expected to pass its version of the reforms by week's end.

"The Senate progress should make it easier for the House to act," Hoyer said.

Rep. Paul Ryan, R-Wis., expressed similar sentiments on CBS "This Morning" earlier in the day. But Ryan added that the House would take up its own immigration legislation and use a more piecemeal approach.

"The House is going to do its own legislation, our own plan which is going to be far more methodical. We're going to take our time," said Ryan. "What the Senate just did is they moved closer to the House's position, which obviously makes final legislation more likely."