The more literal answer to your question is: TeX, the engine behind LaTeX, is Turing-complete so it could in principle do anything you can program in another language. But you'd be crazy to try, because TeX is not a general-purpose programming language.
LaTeX doesn't handle spelling correction either, that's your latex editor at work. In fact TeX cannot control what you see in your editor window-- only the formatted document and auxiliary files it generates. So you should focus on choosing an editor that does what you need. Looks like you use WinEdt, which according to this post had no plans to add grammar checking ten years ago. A minute's googling turned up this page, which suggests TeXstudio is the way to go. Here's also an older tex.sx question which asks the same question, but for Linux. (Since you're on Windows I'm not proposing it as a duplicate, but note that several solutions are available cross-platform.)
This is all based on a quick check, so I suggest you do some more research in case there are options that suit you better.
I should add that nothing keeps you from firing up Word on your LaTeX document and looking for the green lines. It'll get confused by latex mark-up, but for clean runs of text its grammar checker is probably far superior to the alternatives. (Just make sure it's saved in plain-text mode again!)