Gear shifts and drive speed on bikes
For this you will need to know the gearing ratio from front to back (or number of teeth on each gear), the wheel diameter, and the rate at which you are pedaling.

1) RPM to rad/s:
Firstly determine how fast the front gear is turning, assuming that you are pedaling at one complete revolution per 2 seconds: 30 RPM / 60 (convert from minutes to seconds) * 2pi rad/rev:

pedalSpeedRPM = 30
speedRadPerSec = pedalSpeedRPM * 2 * pi / 60


2) Next determine how fast the rear wheel is spinning by using the gear ratio; the ratio of the number of teeth on the front ring to the number of teeth on the back ring. For example, if the front ring has 53 teeth and the rear ring has 13 teeth, then the ratio is 4.08. That means that the rear wheel is spinning 5.2 times faster than the pedals. Thus:

gearRatio = 4.08
speedRearWheel = speedRadPerSec * gearRatio


3) Rotational to linear motion: If you know the diameter of your wheel, you can determine how far it moves with one revolution, for example, a 26" wheel (in metric because that's what I know) has a radius of approximately 0.33m. Using the formula velocity = radius x angular velocity gives

wheelRadius = 0.33
velocity = wheelRadius * speedRearWheel
velocityKPH = velocity * 3.6


So to bring it all together, you need to know the following variables:

pedalSpeedRPM = 30
gearRatio = 4.08
wheelRadius = 0.33


Calculations:
speedRadPerSec = pedalSpeedRPM* 2 * pi / 60
speedRearWheel = speedRadPerSec * gearRatio
velocity = wheelRadius * speedRearWheel
velocityKPH = velocity * 3.6


Running this on MATLAB gives the result:

velocityKPH =

   15.2274


Which isn't a bad estimate for middle gear, non-frantic pedaling.

Another example, with (maybe) more realistic numbers: 

pedalSpeedRPM = 60; % Faster pedalling
gearRatio = 4.2; % Abitrary
wheelRadius = 0.37; % 29"

velocityKPH =

   35.1507


So I guess to (finally) answer your question, yes, you can determine it, but you need to know the gearing ratio, your wheel size, and an estimate of your pedaling rate.