  Massive black hole binaries (MBHBs) form as a consequence of galaxy mergers.
However, it is still unclear whether they typically merge within a Hubble time,
and how accretion may affect their evolution. These questions will be addressed
by pulsar timing arrays (PTAs), which aim to detect the GW background (GWB)
emitted by MBHBs during the last Myrs of inspiral. Here we investigate the
influence of differential accretion on MBHB merger rates, chirp masses and the
resulting GWB spectrum. We evolve a MBHB sample from the Illustris hydrodynamic
cosmological simulation using semi-analytic models and for the first time
self-consistently evolve their masses with binary accretion models. In all
models, MBHBs coalesce with median total masses up to $1.5 \times 10^8
M_{\odot}$, up to $3-4$ times larger than in models neglecting accretion. In
our model with the largest plausible impact, the median mass ratio of
coalescing MBHBs increases by a factor $3.6$, the coalescence rate by $52.3\%$,
and the GWB amplitude by a factor $4.0$, yielding a dimensionless GWB strain
$A_{yr^{-1}} = 1 \times 10^{-15}$. Our model that favours accretion onto the
primary MBH reduces the median mass ratio of coalescing MBHBs by a factor of
$2.9$, and yields a GWB amplitude $A_{yr^{-1}} = 3.1 \times 10^{-16}$. This is
nearly indistinguishable from our model neglecting accretion, despite higher
MBHB masses at coalescence. \textbf{We further predict binary separation and
mass ratio distributions of stalled MBHBs in the low-redshift universe, and
find that these depend sensitively on binary accretion models. This presents
the potential for combined EM and GW observational constraints on merger rates
and accretion models of MBHB populations.}
