"Some philosophers and scientists have argued that there is no scientific method; they include physicist Lee Smolin and philosopher Paul Feyerabend (in his Against Method)"
And even the previous paragraphs are not complete either. It says there:
"Scientists then test hypotheses by conducting experiments or studies. A scientific hypothesis must be falsifiable"
Which is not necessarily true - for example, many arguments against the theory of evolution question whether it is a falsifiable theory (which is not saying that it is not true, but if there is no way to falsify it, how can it be said it is a scientific theory?). Many fields within Biology also have no predictive, but only descriptive power.
Scientists who stick to a rigorous scientific method, such as seen in physics would have to reject most other branches of science. For a realist to accept scientific results in multiple fields, they have to use some philosophical reasons to accept the differing methods as valid ways of exploring reality.