I have never heard of "due diligence" being applied to a simple house purchase, though I suppose it could be introduced in a large and complex sale of an estate containing many houses.
"Due Diligence" is essentially as you described in the US, however the term has been unformalized to be used in many life situations.
For example, long before The Vet and Pedro coupled up, you might have overheard him say... "I should have done my due diligence before I shagged that one".