Not quite. From the doppler-shifted spectrosocopy lines, we can figure out what speed everything is moving relative to us, but that by itself doesn't tell us anything other than their speed.
Noting that all the doppler shifts of other galaxies are red-shifts tells us that all the galaxies are moving away from us, but it still doesn't tell us whether speed correlates to distance in any way.
Figuring out the distance of galaxies is a different matter entirely, and is done with standard candles[0].
It's from the standard candles that we can measure distance, and only once we know the distance that way can we make a correlation between speed and distance, from which we see that the further away a galaxy is, the faster it is receding.
Only once we have found that there is a speed/distance curve can we estimate the distance to an object based on its doppler shift. We can't get it from the doppler shift alone.
Noting that all the doppler shifts of other galaxies are red-shifts tells us that all the galaxies are moving away from us, but it still doesn't tell us whether speed correlates to distance in any way.
Figuring out the distance of galaxies is a different matter entirely, and is done with standard candles[0].
It's from the standard candles that we can measure distance, and only once we know the distance that way can we make a correlation between speed and distance, from which we see that the further away a galaxy is, the faster it is receding.
Only once we have found that there is a speed/distance curve can we estimate the distance to an object based on its doppler shift. We can't get it from the doppler shift alone.
[0] http://en.wikipedia.org/wiki/Cosmic_distance_ladder#Standard...