Unbelievable player, but looking at the numbers, I think it would be an incredibly remote chance for him to catch Rose.
He has 1708 hits and is 34 (turns 35 in October), assume 100 more hits this year = 1800. He was 27 in 2001, his rookie year. Say he came up at 22 and averaged 227 hits per year (his career average) for the 5 years in question (Jeter came up at age 22 and had 1008 hits in his first 5 years), that is 2935 at the end of this season meaning he is still ~1300 short. He would have to maintain his career average of 227 for 6 more years (ages 35-40) to get there.
He is in great shape and a remarkable hitter, but AVERAGING 227 hits over 19 years is asking a bit much. There have only been 50 227 hit seasons in the history of baseball. He would probably have had to play full seasons from age 22 through 43 to do it.