The intrinsic metric induced by a complete metric space is always a complete extended metric. To prove this, we shall write $d^{\sharp}$ for the extended metric $dist(x,y)=\inf(L(\gamma))$.
Also, I want to remark that $d^{\sharp}$ is in general not a metric since we may have $d^{\sharp}(x,y)=\infty$ even when $X$ is path-connected. For instance, if we take some fractal $X$ such the boundary of the Koch snowflake, then $dist(x,y)=\infty$ for every pair of points $x,y\in X$.
However, $d^{\sharp}$ is always an extended metric (By an extended metric I mean that $d^{\sharp}$ is a metric in every way except for the fact that we may have $d(x,y)=\infty$). Notions such as Cauchy sequences and completeness still hold if we use extended metrics instead of metrics. Furthermore, if we let $d'(x,y)=Min(d^{\sharp}(x,y),1)$, then $d'$ is a metric with the same uniform structure as $d^{\sharp}$(and hence the same Cauchy sequences, topology,...).
We shall now prove that $d^{\sharp}$ is complete using standard arguments.
To prove that $d^{\sharp}$ is complete assume that $(x_{n})_n$ is a Cauchy sequence with respect to the extended metric $d^{\sharp}$. Then since $d(x,y)\leq d^{\sharp}(x,y)$ for each $x,y\in X$, the sequence $(x_{n})_{n}$ is Cauchy with respect to the metric $d$. Since $(X,d)$ is a complete metric space the sequence $(x_{n})_{n}$ converges to some point $x\in X$ with respect the metric $d$. Since $(x_{n})_{n}$ is Cauchy with respect to $d^{\sharp}$, we may take a subsequence $(y_{n})_{n}$ of $(x_{n})_{n}$ such that $d^{\sharp}(y_{n},y_{n+1})<\frac{1}{2^{n}}$ for all $n$. In this case, there is some path $\gamma_{n}:[0,1]\rightarrow X$ such that $\gamma_{n}(0)=y_{n},\gamma_{n+1}(1)=y_{n+1}$ and
$L(\gamma_{n})<\frac{1}{2^{n}}$. If we let $\gamma:[0,\infty]\rightarrow X$ be the function where $\gamma(r+n)=\gamma_{n}(r)$ for $n\geq 0$ and $r\in[0,1]$ and $\gamma(\infty)=x$, then $\gamma$ is a continuous function. However, the restricted function $\gamma|_{[k,\infty]}$ is a path from the point $y_{k}$ to $x$ with $L(\gamma|_{[k,\infty]})<\sum_{n=k}^{\infty}\frac{1}{2^{k}}=\frac{1}{2^{k-1}}$. Therefore $d^{\sharp}(y_{k},x)<\frac{1}{2^{k-1}}$, so the sequence $(y_{n})_{n}$ also converges to $x$ with respect to the generalized metric $d^{\sharp}$.
