That depends where your second grader is and which school he/she is enrolled in. Some schools and districts do require a test, and some do not. You can find out by calling the school or talking to the board of ed.
From my experience holding a student back is a state-level decision, with some rare exceptions of school districts allowing for this. I know in GA, where I taught, if a child did not pass the reading section of the 3rd grade send of year state test, they had the option of going to summer school and retaking the test. If they did and passed they moved onto to the next grade.
I would check with your district. I would be surprised if a public school made their own rules governing who passes and who stays back based on a test.