Can cancer be diagnosed by a normal primary care physician?

I would expect many to suspect the diagnosis and tell their patient but sent them to a specialty referral person to confirm/exclude/devise treatment. Much depends on the local health setting.

Of course. Skin cancers could to reasonably diagnosed by visual examination , though most doctors would do a biopsy. A primary care doctor may be able to diagnose leukemia by examining a smear of peripheral blood. Many cancers can be tentatively diagnosed by physical examination and imaging studies.