There is a tantalizing reference to a proof of Euclid via Eratosthenes referenced in this paper, reference 139. However, that reference was posted on a now-defunct internet forum, and the Wayback Machine has so far been unhelpful.
There is also a sort-of proof here on math.SE, but it uses the properties of the sieve (density) rather than the sieve itself directly. So I've been a bit stymied finding what I actually want.
I present what I believe to be a proof below, starting after the horizontal rule. I'm uncertain of its validity, however, partly because it's the first time I've attempted something relatively formal like this. I assume it's not a novel proof; I can't find one like it, but I can't imagine this line of reasoning hasn't been attempted before.
Any comments/suggestions/"this is a glaring error, do you even math bro?" for my proof would be welcomed, as would references for previous examples of proof via Eratosthenes.
We can treat the Sieve of Eratosthenes as a stack of separate sieves, each admitting the numbers indivisible by a prime. If we treat each layer as a set, we can define:
$$E_2 = \{x \in \mathbb{N} \mid x>2 \ \land\ 2 \not \mid n \} \\ E_3 = \{x \in \mathbb{N} \mid x>3 \ \land\ 3 \not \mid n \}$$
and so forth. Each of these sets determines which numbers are admitted through the sieve.
If we stack the layers one on top of the other, as the Eratosthenes algorithm does, we get a set of modular[a] sieves, with primorial moduli:
$$E_{3\#} = E_2 \cap E_3 = \{ x \in \mathbb{N} \mid x = 1, 5 \pmod 6\} \\ E_{5\#} = E_2 \cap E_3 \cap E_5 = \{ x \in \mathbb{N} \mid x = 1, 7, 11, 13, 17, 19, 23, 29 \pmod {30} \}$$
and so on. We will call these Eratosthenean sieves, or E-sieves. Note that while the sieves are modular, they do exclude the primes that form them; however, all those primes are already known to be prime, as they were used to construct the sieve.
Each E-sieve has a modulus $M_{p_n\#} = p_1p_2\cdots p_n =p_n\#$. We can also define an admittance[b], that is, the quantity of numbers admitted by the E-sieve over the length of the modulus: $A_{p_n\#} = (p_1-1)(p_2-1)\cdots (p_n-1)$. Finally, we can define a validity ceiling for each E-sieve, that is, the largest number for which the sieve can distinguish between a prime and a composite odd number with certainty: $V_{p_n\#} = (p_n + 1)^2 - 2$. Above $V_{p_n\#}$, at least one more layer of the sieve is needed to be certain $V_{p_n\#}$ is prime (or composite).
For the first few sieves, we have:
$$\begin{array}{|c|c|c|c|} \hline & M_{p_n\#} & A_{p_n\#} & V_{p_n\#} \\ \hline E_{2\#} & 2 & 1 & 7 \\ \hline E_{3\#} & 6 & 2 & 23 \\ \hline E_{5\#} & 30 & 8 & 47 \\ \hline E_{7\#} & 210 & 48 & 119 \\ \hline E_{11\#} & 2310 & 480 & 167 \\ \hline \end{array}$$
$M_{p_n\#}$ and $A_{p_n\#}$ are much, much larger than $p_n$ for even small $p_n$, because they grow faster than exponentials. Chebyshev's first function gives us, asymptotically,
$$ M_{p_n\#} = p_n\# \sim e^{p_n} $$
And using Mertens's third theorem we have
$$A_{p_n\#} = M_{p_n\#} \cdot \prod^{n}_{i}{\left(\frac{p_i - 1}{p_i}\right)} \sim \frac{e^{p_n - \gamma}}{\log p_n} $$
In terms of length, $M_{p_n\#}$ has approximately $p_n \log_{10} e$ (decimal) digits, and $A_{p_n\#}$ has approximately one fewer digit.[c]
$V_{p_n\#}$ increases as a polynomial, and therefore, even for small $p_n$, $V_{p_n\#}$ <<< $A_{p_n\#}$.
Now we will prove Euclid's Theorem by contradiction.
Theorem (Euclid's Theorem): The set of prime numbers is an infinite set.
Assume the statement is false. This implies that there is some final, largest prime, which we will call $p_f$.
In order to have determined that $p_f$ is prime using E-sieves, we also must have found all the primes smaller than $p_f$. Therefore, we may construct the E-sieve $E_{p_f\#}$ by using all of these primes. This E-sieve must have an admittance many, many orders of magnitude larger than $p_f$--approximately $p_f \log_{10} e$ digits long.
$A_{p_f\#}$, therefore, is larger than $p_f$ itself. Even if every number below $p_f$ were prime, $A_{p_f\#}$ is large enough that the sieve admits numbers greater than $p_f$.
We used all the primes up to and including $p_f$ to construct this sieve. Now, each admitted number larger than $p_f$ must either:
(1) be prime
(2) be composite, and blocked from admission by a sieve larger than $E_{p_f\#}$.
In either case, there must be a prime number larger than $p_f$, contradicting our initial assumption. Therefore, by contradiction, the statement must be true.
[a]I think modular is the correct term here, though I've also seen the modulus called the "width" of the sieve; would that be more proper?
[b]I don't know if there is a standard name in sieve theory for this quantity. I've simply reached into by chemistry background and yoinked this from spectroscopy.
[c]This section could likely refrain from using Chebyshev and Mertens, as neither Euclid nor Eratosthenes would have has access to (or understanding of) those results. In addition, I do not know if the results of Chebyshev and Mertens rely on the infinitude of primes. This could be written without reliance on them: for example, if you know all of the prime numbers up to 100, you can see that $A_{97\#}$ must have at least 20 decimal digits, because each multiplication by a number larger than 10 adds at least one digit. By similar reasoning, $A_{997\#}$ must have at least 306 decimal digits, and so on. (The actual number of digits is larger, of course: 36 and 415, respectively)