Is meditating while on the bike at the gym an effective method of meditation? Walking meditation : http://www.meditationoasis.com/how-to-meditate/simple-meditations/walking-meditation/ is considered as "just as profound as sitting meditation". If we were to extrapolate this idea into a situation where instead of walking you were on a bike, would an expert's opinion consider this meditating on the bike at the gym an effective practice of meditation?
Short Answer: Yes.

Longer Answer:

The Satipatthana Sutra implies this.  Here's a snippet...


  Moreover, when a practitioner walks, he is aware, ‘I am walking.’ When
  he is standing, he is aware, ‘I am standing.’ When he is sitting, he
  is aware, ‘I am sitting.’ When he is lying down, he is aware, ‘I am
  lying down.’ In whatever position his body happens to be, he is aware
  of the position of his body.
  
  Moreover, when the practitioner is going forward or backward, he
  applies full awareness to his going forward or backward. When he looks
  in front or looks behind, bends down or stands up, he also applies
  full awareness to what he is doing. He applies full awareness to
  wearing the sanghati robe or carrying the alms bowl. When he eats or
  drinks, chews, or savors the food, he applies full awareness to all
  this. When passing excrement or urinating, he applies full awareness
  to this. When he walks, stands, lies down, sits, sleeps or wakes up,
  speaks or is silent, he shines his awareness on all this.


The gist of the above is that mindfulness should be applied throughout the day -- including time on the exercise bike.

However, I don't think this should replace formal meditation practice.  Setting aside time for meditation allows you to better control distractions, which in turn can allow you to go deeper. Meditating on the exercise bike should be done in addition to your regular meditative practice, and should be part of an attempt to maintain continuous mindfulness.