The best thing I have read on this is the chapter, "Fasting for the Little Ones: Abortion and the Sovereignty of God over False Worldviews," from John Piper's Hunger for God (the link takes you to a free PDF of the whole book).
Piper looks at Francis Schaeffer's legacy of worldview engagement, and then writes:
But I wonder if many of the young scholars and activists (now in their forties and fifties!) whom he inspired need to hear a balancing word about the power of prayer and fasting, not as an alternative to thinking and acting, but as a radical foundation that says, “The victory belongs to the Lord, even if the horse (of scholarship and politics) is made ready for the day of battle” (see Proverbs 21:31). Listen to the books crying out for evangelical renewal and reformation in the life of the mind, the restoration of Truth in the place of technique, the recovery of church social compassion from government powerlessness, the taking of moral high ground in the environmental cause, and many other causes. Is there a sense in each of these that the root issues are so intractable to human suasion that the call for fasting and prayer would not only be fitting but desperately needed? I am commending such a call.So what can you pray? Here's one way:
Fasting comes in alongside prayer with all its hunger for God and says,Piper also writes:“We are not able in ourselves to win this battle. We are not able to change hearts or minds. We are not able to change worldviews and transform culture and save 1.6 million children. We are not able to reform the judiciary or embolden the legislature or mobilize the slumbering population. We are not able to heal the endless wounds of godless ideologies and their bloody deeds. But, O God, you are able! And we turn from reliance on ourselves to you. And we cry out to you and plead that for the sake of your name, and for the sake of your glory, and for the advancement of your saving purpose in the world, and for the demonstration of your wisdom and your power and your authority over all things, and for the sway of your Truth and the relief of the poor and the helpless, act, O God. This much we hunger for the revelation of your power. With all our thinking and all our writing and all our doing, we pray and we fast. Come. Manifest your glory.”
I appeal to you to seek the Lord with me concerning the place of fasting and prayer in breaking through the darkened mind that engulfs the modern world, in regard to abortion and a hundred other ills. This is not a call for a collective tantrum that screams at the bad people, “Give me back my country.” It is a call to aliens and exiles in the earth, whose citizenship is in heaven and who await the appearance of their King, to “do business” until he comes (Luke 19:13). And the great business of the Christian is to “do all to the glory of God” (1 Corinthians 10:31), and to pray that God’s name be hallowed and his kingdom come and his will be done in the earth (Matthew 6:9-10). And to yearn and work and pray and fast not only for the final revelation of the Son of Man, but in the meantime, for the demonstration of his Spirit and power in the reaching of every people, and the rescuing of the perishing, and the purifying of the church, and the putting right of as many wrongs as God will grant.I join Piper in commending this practice to you--utterly foolish to the world, but pleasing to God.