Yes and no... Some Ph.D. and Psy.D. programs explicitly require an undergraduate psychology major. Some reputable programs will still consider you if you did not major in psych but have a number of essential prerequisites or a certain number of credit hours (often between 12 and 24). On that note, they'll usually ask for the following required courses:
Introductory psychology;
Research Methods or Experimental Psychology;
Statistics;
Abnormal psychology;
Many will also require a course on the Biological Bases of Psychology/Neuropsychology, cognitive psychology; developmental psychology; personality psychology; and/or social psychology.
I hope that's helpful... Note that these programs can be quite competitive, and the majority of applicants will have been psychology majors. Still, a number of those prereq courses plus solid research and clinical experience may be enough.