I have known a few missionaries who have kept the fast along with the Muslims in order to identify with them. I know that there are objections to such a thing, but we must also realize that there is a matter of one's conscience. Jesus himself talks about fasting:
 “And when you fast, do not look gloomy like the hypocrites, for they disfigure their faces that their fasting may be seen by others. Truly, I say to you, they have received their reward.  But when you fast, anoint your head and wash your face,  that your fasting may not be seen by others but by your Father who is in secret. And your Father who sees in secret will reward you.
(Matthew 6:16-18 ESV)
The Muslims, however, have made a big show of fasting in order to show their piety. This is perhaps one of the issues, I typically have with even the way fasting maybe presented in today's Christian church, such as corporate fasting and the like. Are we putting on a show too, at times?
So, how do we respond to fasting Muslims with the Gospel. We can do as Dr. Phil Parshall (former missionary to Bangladesh and the Phillipines, and acclaimed Islamic scholar) as done on several occasions and keep the fast with them. Not as a show of being holy, but as identifying with them and loving them, all the while following Jesus' guidelines as listed above. One method often used by missionaries is to do a juice and water fast. Basically, you drink juice and water during that time. Another option might be just to offer to pray for the Muslims that you may come in contact with, which we should do anyway. Pray that God would enlighten them to the truth of His Word, that is Jesus Christ. They often ask for guidance, good health, and long life. We should never try to make ourselves appear holier than others, but we should always be consistent in our walk with Christ. They believe that we are hypocrites due to what they see on American television. Let us show them otherwise.