Conservative intellectual and Princeton Professor Robert George points out that liberals are not relativists at all. Rather, they are moralistic dogmatists:
Contemporary left liberals are hardly relativists! I often wish they were. They are moralists—moralists on a mission. The mission is to shape political and social life, and, to the extent possible, individual belief, in line with their passionately held moral convictions. One sees this everywhere, beginning with the war waged by the Obama administration on the Catholic Church—the largest and most important institution whose moral teachings stand in conflict with left liberal beliefs about the status of nascent human life, the nature and meaning of marriage, and religious liberty.
These are not people who deny that there are moral truths. On the contrary, these are people who affirm that there are moral truths and are so certain that they understand them correctly that they are willing to impose them on society. What are some of those “truths”? The absolute right to abortion. The right to conduct one’s sexual life however one pleases, so long as one refrains from obtaining sex by coercion or deception. The conviction that “marriage” is the union of two people (or more) without regard to gender. The idea that the state legitimately may and even should use its coercive powers to prohibit whatever counts in liberal ideology as a form of discrimination. The idea that anyone who disagrees with them about the things they most care about is a “bigot.” And on and on.
As for liberal claims that science is “on their side,” the aim of Conscience and Its Enemies is to show why that can only be regarded as laughable. Consider the unwillingness of so many liberals to face up to the undeniable fact that abortion takes the life of a living human being—a fact established not by theological reflection or religious authority but by modern human embryology and developmental biology.
I would say the left invokes relativism as a way to argue against conservative beliefs, but, yes, they are not relativist when it comes to their own dogmas.