No. Composting doesn't remove the hazardous pathogens from pet waste, and can contaminate the rest of your compost pile.
Most home compost piles don't reach temperatures that are hot enough to kill the hazardous pathogens. Killing E. coli and Salmonella requires extended exposure at 140-degree temperatures. Giardia can survive temperature extremes, chlorination, and drying. Cryptosporidium, Leptospira, Salmonella, and E. coli can all survive for months in feces or soil, and roundworms can survive for up to four years in soil.
Most commercial compost processors also don't reach a temperature high enough to kill the hazardous pathogens, and they don't accept pet waste because it can contaminate the rest of the composted material. So you should keep pet waste out of yard waste bins, too.
The best place for pet waste is in the landfill.