Summary of problems with claim:
The link between evolution on short timescales and longer-term evolutionary processes is thoroughly testable and not a mere extrapolation.
Anyone who denies the logical link between genetic changes within a population ("microevolution") and speciation ("macroevolution") is similar to someone who watches the sun come up in the east and move west across the sky, but denies that it will set in the west. The only difference between genetic changes within a population and generation of a new species from that population is time. Given enough time, the sun will set in the west. Given enough time, speciation will occur.
This claim is related to the "young earth" creationist belief that the earth is only a few thousand years old. In this belief system, there has not been enough time for speciation to occur, given the rate of change that we can observe in most populations. So it is necessary for them to deny reality (observations of speciation) in order to validate a creationist perspective on the age of the earth. An age, by the way, that is about 0.00000002% of the approximately 3 billion years over which biological evolution has proceeded.