Barry Cipra has already computed the first few values. The next couple of numbers $p_n$ are

$13 = 5 \cdot 11 - 2 \cdot 3 \cdot 7$,

$17 = 2 \cdot 7 \cdot 13 - 3 \cdot 5 \cdot 11$,

$19 = 2^2 \cdot 3 \cdot 5 \cdot 17 - 7 \cdot 11 \cdot 13 $,

$23 = 7 \cdot 11 \cdot 17^2 - 2 \cdot 3^2 \cdot 5 \cdot 13 \cdot 19$,

$29 = 3 \cdot 11 \cdot 13^2 \cdot 19 \cdot 23 - 2^{12} \cdot 5 \cdot 7 \cdot 17$.

I don't find such expression for 31 in numbers $\leq 10^{12}$.

The smallest I find is $47 = 3 \cdot 7 \cdot 19^2 \cdot 23 \cdot 29 - 2^5 \cdot 5 \cdot 11 \cdot 13^2 \cdot 17$.

If the abc conjecture is true, there are at most finitely many ways to express the $n$-th prime as a difference of coprime numbers which are divisible only by the first $n-1$ primes.

Barry Cipra has already computed the first few values. The next couple of numbers $p_n$ are

$13 = 5 \cdot 11 - 2 \cdot 3 \cdot 7$,

$17 = 2 \cdot 7 \cdot 13 - 3 \cdot 5 \cdot 11$,

$19 = 2^2 \cdot 3 \cdot 5 \cdot 17 - 7 \cdot 11 \cdot 13 $,

$23 = 7 \cdot 11 \cdot 17^2 - 2 \cdot 3^2 \cdot 5 \cdot 13 \cdot 19$,

$29 = 3 \cdot 11 \cdot 13^2 \cdot 19 \cdot 23 - 2^{12} \cdot 5 \cdot 7 \cdot 17$.

I don't find such expression for 31 in numbers $\leq 10^{12}$.

The smallest I find is $47 = 3 \cdot 7 \cdot 19^2 \cdot 23 \cdot 29 - 2^5 \cdot 5 \cdot 11 \cdot 13^2 \cdot 17$.

If the abc conjecture is true, there are at most finitely many ways to express the $n$-th prime as a difference of coprime numbers which are divisible only by the first $n-1$ primes.

Addendum: For a more extensive table, see http://www.fermatquotient.com/DiverseMinimas/S=M-N_min (found by Google'ing for the numbers obtained above). The data supports the assumption that the OP's assertion is unlikely to be true.