The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a stereographic projection onto $\mathbb{S}^N\setminus\{P\}$. (Here $P$ is either the north or south pole of the unit sphere $\mathbb{S}^N$.) Namely, letting $$ \Omega=\frac{2}{1+\lvert x \rvert^2},\qquad \left(x\in \mathbb{R}^N\right) $$ denote the conformal factor of the stereographic projection, one has the formula $$\tag{1} -\Delta_{\mathbb{R}^N}\phi = \Omega^{\frac{2+N}{2}} \left(-\Delta_{\mathbb{S}^N} + \frac{N(N-2)}{4}\right)\left(\Omega^{1-\frac{N}{2}}\phi\right). $$ My question is: does there exist an analogue of (1) for the fractional Laplacian $(-\Delta_{\mathbb{R}^N})^\alpha$? Actually I am interested in the cases $\alpha=\pm \frac{1}{2}$.

Unless I'm misreading, there exist such an analogue at least for some $\gamma$. See equation 2.5 and theorem 3.2 of Fractional Laplacian in Conformal Geometry by Sun-Yung Alice Chang and María del Mar Gonzáles.

To spell it out, the theorem 3.2 says that for $\gamma \in (0,\frac{n}{2})\setminus \mathbb{N}$ and a smooth function $f:\mathbb{R}^n\to \mathbb{R}$ one has $$ P_\gamma[g_\mathbb{H},|dx^2|] f = (-\Delta_x)^\gamma f, $$ where $P_\gamma[g^+,\widehat{g}]$ is the fractional power of the conformal Laplacian with power $\gamma$ associated to the ambient metric $g^+$ and the conformal class $[\widehat{g}]$. The metric $g_\mathbb{H}$ is the standard hyperbolic metric of the upper space and $|dx^2|$ is the standard Euclidean metric.

Now if we take the rescaled metric $\widehat{g}_v = v^{\frac{4}{n-2\gamma}}\widehat{g}$, equation 2.5 claims that $$ P_\gamma[g^+,\widehat{g}_v] \phi = v^{-\frac{n+2\gamma}{n-2\gamma}}P_\gamma[g^+,\widehat{g}] (v \phi), $$ for all smooth functions $\phi$.

The conformal fractional Laplacian $P_\gamma[g_\mathbb{H},v^{\frac{4}{n-2\gamma}}|dx^2|]$ plays the role of the spherical conformal Laplacian in the classical formula (1) of OP.

Just a short addendum: long ago Stefan Samko observed this connection between Euclidean and spherical fractional Laplacians, see item d) in page 360 of his conference paper from 2000.