No, this is not true.  Let $G$ be the bowtie graph (this is the graph obtained by gluing two triangles at a vertex $u$).  Then, $G$ does not have a spanning regular subgraph, but $\mathbb{1}$ is in the row space of $G$.  Just set $x_e=\frac{1}{4}$ if $e$ is adjacent to $u$ and $x_e=\frac{3}{4}$ for the other two edges.