Obviously a star would be a point source. A galaxy should be an irregular blob if close, but if it is far away then it would seem that a galaxy too would be just a point source.
Given that the star and galaxy were both only detectable as point sources can astronomers tell them apart with redshift? By some other method?
A followup question...
What percentage of galaxies if our universe can we only detect as point sources?
To distinguish galaxies from stars, you can use the spectrum. Roughly, stars have a black-body like spectrum with features depending on the absorption and emission on the line of sight and in the chromosphere of the star.
Galaxies on the other hand of a spectrum that is the composite of tons of stars. The spectrum will for example be much wider (ranging from smaller to larger wavelengths) because of the diversity in the spectra of stars.
Take a look at http://www.atnf.csiro.au/outreach/education/senior/astrophysics/spectra_astro_types.html if you want a quick overview of the differences.
I don't have a precise number about the number of galaxies we see as point source, but the answer varies greatly from one instrument to another one. If you try to observe a galaxy using radiotelescopes in interferometry, you can resolve much better scales than an Earth-based small visible telescope, etc…
cphyc's answers the question excellently: Spectroscopy is the answer, although since — as explained below — galaxies are not point sources, the morphology of stars and galaxies is also different: even elliptical galaxies observed along one of their axes look different from stars. Although both are round, the way that their light falls off radially is different; stars' light decrease roughly as a normal distribution from the center and out (with some extra profile folded in which depends on the instrument), while the surface brightness profile of galaxies decrease in a somewhat more complicated fashion (e.g. a Sérsic profile).
Can galaxies be point sources?Wrt. the fraction of galaxies that are point source, the answer is virtually none. Galaxies can almost always be resolved although, as cphyc also correctly says, not with any instrument. Radio and gamma-ray telescopes have very poor resolution, and at these wavelengths the sources usually cannot be resolved unless they're relatively nearby. But at optical wavelengths, as well as UV and IR, telescopes like the Hubble Space Telescope and even good ground-based telescopes can resolve ~all galaxies, unless they're so small that they're too dim to be seen anyway.
Angular diameter in an expanding UniverseThe reason is a rather peculiar feature of the expanding Universe: A galaxy will look smaller and smaller, the farther away it is (as expected from everyday life), but only out to a certain distance, after which they will appear larger and larger. Why is this so? Because light moves with a finite speed, we observe galaxies as they were in the past — the more distant, the longer time ago. And since in an expanding Universe, "long time ago" also means closer, the angle that a galaxy spans on the sky is the angle that it spanned when it emitted the light, not the angle it spans today. That is, very distant galaxies emitted the light we see today when they were so close that they spanned a large angle.
The exact relation between distance and the solid angle of a galaxy depends on the cosmology (i.e. the values of density parameters, Hubble constant, etc.). For the latest Planck measurements (2015), a galaxy that is 1 kpc (~3000 lightyears) across — which would be considered a small galaxy — spans an angle given by this figure:
You'll see that galaxies look smaller and smaller the farther there are away, until at a distance of roughly 15 billion lightyears, after which they look larger again. The most distant galaxy observed, GN-z11, is so far away that its light was emitted less than half a billion years after the Big Bang. With a radius of $0.6\pm0.3\,\mathrm{kpc}$ (Oesch et al. 2016) it still spans 0.15 arcsec, which is resolvable by HST.
Decreasing surface brightnessUnfortunately, this effect also makes distant galaxies more difficult to detect. A galaxy only emits so much light, so distributing its light over, say, twice the angular diameter, makes it four times less bright.
Thus, the problem of observing very distant galaxies is not that they're small, but that they're dim.
Good answers have already been given, but I wanted to provide another way of looking at it. Take a look at the image below, which is the Hubble Extreme Deep Field (XDF) $-$ for those who don't know, this is a small patch of the sky that Hubble has stared at for a total of 23 days over 10 years $-$ and you'll notice something interesting. It is clear to see that many of the larger objects are galaxies, but you'll see a great number of smaller pinpoints of light (nearly 5,500 of them) which are galaxies so far away that Hubble can barely resolve their extent and size. Now take a look at the bright object in the lower right quadrant. You should see that it has blue and red spikes around it, referred to as diffraction spikes. This object is clearly a star and you can primarily tell because of the diffraction spikes. You don't see these diffraction spikes on the galaxies, even the galaxies that are tiny pinpoints. This is a relatively easy way to visually distinguish between a star and a galaxy when you're looking at it through a telescope where such diffraction spikes are expected to occur.
This implies that visually, stars and galaxies look different, even if they're both tiny spots on the image. There will also be differences in the way they look in less perceptible ways. This concept is capitalized upon by a program widely used by astronomers, SExtractor, designed to be given an image of the sky and be able to differentiate between stars and galaxies. It uses these small differences between the ways galaxies and stars appear in images to figure out which is which. If you want more detailed information on how this program distinguishes between stars and galaxies, take a look at their published paper.
