First of all, note that this value can be achieved with the following fractions choice:
$$\frac{1}{1011}, \frac{2}{1012}, \frac{3}{1013}, \dots, \frac{1010}{2019}.$$

For the sake of contradiction, let's suppose that the smaller value of the maximum fraction can be obtained in some other way. Clearly, $2019$ cannot be a numerator, hence we can consider the fraction with denominator $2019$. Its numerator should be smaller than $1010$. Let's denote it as $a < 1010$.
There are at least $1009$ numbers in the set
$$M = \{a, a+1, \dots, 1010, 1011, \dots, 2018\}.$$
From the pigeonhole principle it follows that some of these numbers form one of the rest $1008$ (excluding $\frac{a}{2019}$) fractions. If we denote them $b < c$ then we will have $\frac{1010}{2019} < \frac{b}{2019} < \frac{b}{c}$, a contradiction.