Republican Rep. Paul Ryan unveiled a new Medicare proposal Thursday that would give future seniors the choice of purchasing private insurance coverage or staying in the traditional federal plan.

The concept, which is backed by Democratic Sen. Ron Wyden of Oregon, steps back from the House budget chairman's previous plan to end the traditional fee-for-service Medicare program for future retirees and replace it with subsidies starting at $8,000 that seniors would use to purchase private health plans.

That...