At the end of the first mile, there will be $ \frac{2}{3} $ of the initial milk in the bucket. Each additional mile multiplies this amount by $ \frac{2}{3} $. Thus, when he arrives at home at the end of the third mile, there will be $ \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \left(\frac{2}{3}\right)^{3} $ as much milk in the bucket. Since he had 2 gallons initially, the amount in the bucket when he gets home is $ 2 \cdot \left(\frac{2}{3}\right)^{3} $. Because $ \left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}} $, this expression is equivalent to $ 2 \cdot \frac{2^{3}}{3^{3}} $. Because $ n^{a} \cdot n^{b} = n^{a+b} $, this equals $ \frac{2^{4}}{3^{3}} $. Multiplying the exponents out, we get $ \boxed{\frac{16}{27}} $ gallons.