No.
The concept behind unit tests is based on a premise that has been known to be false since before unit testing was ever invented: the idea that tests can prove that your code is correct.
Having lots of tests that all pass proves one thing and one thing only: that you have lots of tests which all pass. It does not prove that what the tests are testing matches the spec. It does not prove that your code is free from errors that you never considered when you wrote the tests. (And the things that you thought to test were the possible issues you were focusing on, so you're likely to have gotten them right anyway!) And last but not least, it does not prove that the tests, which are code themselves, are free from bugs. (Follow that last one to its logical conclusion and you end up with turtles all the way down.)
Djikstra trashed the concept of tests-as-proof-of-correctness way back in 1988, and what he wrote remains just as valid today:
It is now two decades since it was pointed out that program testing may convincingly demonstrate the presence of bugs, but can never demonstrate their absence. After quoting this well-publicized remark devoutly, the software engineer returns to the order of the day and continues to refine his testing strategies, just like the alchemist of yore, who continued to refine his chrysocosmic purifications.
The other problem with unit testing is that it creates tight coupling between your code and the test suite. When you change code, you'd expect some bugs to show up that would break some tests. But if you're changing code because the requirements themselves have changed, you'll get a lot of failing tests, and you'll have to manually go over each one and decide whether or not the test is still valid. (And it's also possible, though less common, that an existing test that should be invalid will still pass because you forgot to change something that needed to be changed.)
Unit testing is just the latest in a long line of development fads that promise to make it easier to write working code without actually being a good programmer. None of them have ever managed to deliver on their promise, and neither does this one. There is simply no shortcut for actually knowing how to write working code.