Can you lose weight by walking one mile a day?
You're asking the wrong question.
Exercise is perhaps the single best thing you can do for your health, but exercise is not a weight-loss tool. Especially when we're talking about walking for a mile.
A quick Google search shows that a mile walk burns around 100 calories. Similar estimates are all over the internet. This means you'd burn less than half of the calories found in a single Cliff Bar.
Weight loss and weight gain begin in the kitchen. If you want to shed some pounds, then focus on you're putting into your body.
Use a program like MyFitnessPal to get a rough estimate of how much you should eat, enter your target weight and time frame, and stick to that meticulously. Input everything you eat. Do it for a few months to build some good habits and enable yourself to do the mental calculations without relying on the program. It's not about eating less, but rather eating the right amount.
And keep it simple. The perfect diet is the one about which you can say, "I'd feel comfortable eating like this for the rest of my life". There is no One True Diet. People will argue endlessly about this, but it's counterproductive. Forget all of the bullshit books you'll find at Barnes and Noble--"bulletproof" this and low-carb that. It's a waste of your time and money. Almost all of them work if you actually do what they say, but few offer practical, sustainable, long-term solutions.
I used to weigh eighty pounds heavier, so I'm telling you this from experience. I exercise a lot right now since I'm an active runner and cyclist, but that's not the reason I lost weight. There's plenty of physically fit obese folks.
Now some will disagree and say that exercise does help you lose weight. There's some truth to that. If you're cycling for twenty to forty miles at a time, four days a week, then you're going to burn a lot of calories. Distance runners know this too. Running upwards of twenty or thirty miles a week burns a lot of calories. But what most people consider "working out" can be offset by one sports drink or bite-sized candy bar. And what do most people do immediately after they exercise? They eat!
So get your diet under control.
But go outside and walk anyway. It's a peaceful, contemplative activity that's good for your physical and mental health. Couple that with positive dietary changes, and you'll be off to a great start!