  Galactic core-collapse supernovae are among the possible sources of
gravitational waves. We investigate the ability of gravitational-wave
observatories to extract the properties of the collapsing progenitor from the
gravitational waves radiated. We use simulations of supernovae that explore a
variety of progenitor core rotation rates and nuclear equations of state and
examine the ability of current and future observatories to determine these
properties using gravitational-wave parameter estimation. We use principal
component analysis of the simulation catalog to determine the dominant features
of the waveforms and create a map between the measured properties of the
waveform and the physical properties of the progenitor. We use Bayesian
parameter inference and the parameter map to calculate posterior probabilities
for the physical properties given a gravitational-wave observation. We estimate
the ratio of the progenitor's core rotational kinetic energy to potential
energy ($\beta$) and the post bounce oscillation frequency. For a supernovae at
the distance of the galactic center (8.1 kpc) with $\beta = 0.02$ our method
can estimate $\beta$ with a $90\%$ credible interval of $0.004$ for Advanced
LIGO, improving to $0.0008$ for Cosmic Explorer. We demonstrate that if the
core is rotating sufficiently rapidly for a signal observed by Cosmic Explorer,
our method can also extract the post bounce oscillation frequency of the
protoneutron star to a precision of within $5$~Hz ($90\%$ credible interval)
allowing us to constrain the nuclear equation of state. For a supernovae at the
distance of the Magellanic Clouds (48.5 kpc) Cosmic Explorer's ability to
measure these parameters decreases slightly to $0.003$ for rotation and $11$~Hz
for the postbounce oscillation frequency ($90\%$ credible interval). Sources in
Magellanic Clouds will be too distant for Advanced LIGO to measure these
properties.
