Welp, you need to check almost all rows and columns.
Proof by counterexample:
Let X be a correct solution. Switch the topleft square with its right neighbour. All rows and boxes still check out but two columns do not. Same for rows if you switch two vertically aligned squares. So you can skip at most one row/column.
The only thing you can skip, after having checked all 9 of one item(say rows), is that you only need to check 8 of the other 2 items(columns and boxes), as it's already clear that there are 9 of each number. Alternatively, after fully checking rows and columns, you can skip boxes 1, 5 and 9(or 3 other boxes so that they do not align)(because there are 3 of the number in the 3 rows/columns it uses, and the other 2 boxes in those rows/columns are checked).
EDIT: That's a mighty useful link, mathoverflow has thought this through a lot. Paraphrasing the pages of text found there:
When you have checked the columns and rows, you can skip some boxes. Having checked the top 3 rows and boxes 1 and 2, you don't need to check box 3. Likewise you don't need to check box 6 if you have 4 and 5, and the bottom boxes are proven correct by having checked the columns and the first six boxes.
The last row is proven correct by the bottom 3 boxes and the two rows above it. So you can skip 1 row and 5 boxes for 21 checks total.