Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable

As it has been pointed out by Ramiro, the assertion to be proved in the question is false. Only one of the implications is correct that is if $(\Sigma, \rho_{\Delta})$ is separable, then $L^p(X,\Sigma, \mu)$ is separable. This is what will be shown below.

Let $\chi_A$ be the characteristic function of the set $A$. Now observe that

$\| \chi_A - \chi_B\|_{L^p}^p = \int|\chi_A - \chi_B|^p = \int_{A\setminus B}1 + \int_{B\setminus A}1 = \mu(A\setminus B) + \mu(B\setminus A) = \rho_{\triangle}(A,B)$

Now consider the function $\varphi: \Sigma \to L^p(X, \Sigma,\mu )$ defined as $ A \mapsto \chi_A$.

This is an injective map from $(\Sigma, \rho_{\triangle})$ to $(L^p(X, \Sigma,\mu), \rho_p)$.

Observe that $\rho_{\triangle}(A_i,B) \rightarrow 0 \Leftrightarrow \|\chi_{A_i} - \chi_{B} \|_{L^p} \rightarrow 0$.

Consider a countable dense subset in $L^p$ say $f_i$. Now choose an element from each of the following sets $B(f_i,1/k)\cap \mathrm{Im}(\varphi)$ say $\chi_{A_{i,k}}$ where $i$ and $k$ vary over natural numbers. This will be a countable subset of $\mathrm{Im}(\varphi)$. For any given $\varepsilon$ and $\chi_A$ there exists $f_j$ such that $\| f_j - \chi_A \|_{L^p} < 1/k < \varepsilon/2$. Now let's choose $\chi_{A_{j,2k}}$, then $\| \chi_{A_{j,2k}} - \chi_A \|_{L^p} \leq \varepsilon $. Mistake in this argument was also pointed out by Glinka below.

Let $\lbrace A_i \rbrace$ to be a dense subset in $\Sigma$. Now consider the vector space spanned by $\mathbb{Q}\chi_{A_i}$. This is a countable set. This is dense in the vector space spanned by $\mathbb{R}\chi_{A_i}$ which is dense in $L^p$ since simple functions are dense in $L^p$. One can suitably modify the proof for complex valued functions.