Not 100%. Amtrak’s FY2022 fact sheet says that 72.6% of miles driven was on tracks owned by other railroads. That doesn’t tell us how much rail they own, but they do own some of their own rail.
Not only does it mean higher prices, but worse service because they have to rely on freight companies to maintain their rails properly and preference is given to freight leading to passenger delays.
Edit: Found a better source.
Ninety-seven percent of the route-miles traveled by Amtrak trains are on tracks owned by other railroads.