Could I lose weight and improve fitness simply by healthy eating without exercise?

Minimal. Make sure you exercise at least 30 minutes a day, don't skip meals, avoid soda, juice, dairy, alcohol and fried foods. Eat a healthy well balanced diet! also water, organic vegetable broth, green leafy vegetables, natural pickles, apples are great to fill you up so you dont feel hungry and they are very low in calories.
Yes, but.... You can lose weight and improve fitness through healthy eating only, but, one loses all the benefits of exercise. This includes, better heart health, cholesterol, blood sugar, muscle tone, mental status, etc. Both exercise and healthy eating go hand-in-hand, with exercise being equally, if not more important.