Under a weak prior $\pi (\theta) \propto  1$ the posterior is $\pi (\theta | x ) \propto \mathcal{L} (\theta)$ and then the normalising constant is just $\int \mathcal{L}(\theta) d\theta$. Hence, the posterior expectation is 
\begin{align*}
E(\theta|x) & = \int \theta \pi(\theta | x) d\theta \\
& = \int \frac{\theta \mathcal{L}(\theta)}{\int \mathcal{L}(\theta) d\theta} d\theta \\
& = \frac{\int \theta \mathcal{L}(\theta) d\theta}{\int \mathcal{L}(\theta) d\theta}
\end{align*}

So $\hat{\theta}$ is can be interpreted as the posterior expectation of $\theta$ under the improper prior $\pi(\theta) \propto 1$. [Wikipedia](https://en.wikipedia.org/wiki/Bayes_estimator#Generalized_Bayes_estimators) tells me this is the ''generalised Bayes estimator''. Unfortunately, I don't know much about the generalised bayes estimator but know that we know it's a thing then someone else on stackexchange can help us find out more about it!