Step 1: Find the slopes of the given lines. For $y = 3x + 1$, slope $m_1 = 3$. For $2y = x + 3$, rewriting as $y = \frac{1}{2}x + \frac{3}{2}$, slope $m_2 = \frac{1}{2}$.

Step 2: Apply the formula for angle between two lines. The acute angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by $\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$.

Step 3: Set up the equation for equal inclination. For the line $y = mx + 4$ to be equally inclined to both lines: $\left| \frac{m - 3}{1 + 3m} \right| = \left| \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} \right| = \left| \frac{2m - 1}{2 + m} \right|$.

Step 4: Solve the equation. This gives two cases: $\frac{m - 3}{1 + 3m} = \frac{2m - 1}{2 + m}$ (no real solution) or $\frac{m - 3}{1 + 3m} = -\frac{2m - 1}{2 + m}$.

Step 5: For the second case, cross-multiply: $(m - 3)(2 + m) = -(2m - 1)(1 + 3m)$, which simplifies to $7m^2 - 2m - 7 = 0$.

Step 6: Apply the quadratic formula: $m = \frac{2 \pm \sqrt{4 + 196}}{14} = \frac{2 \pm 10\sqrt{2}}{14} = \frac{1 \pm 5\sqrt{2}}{7}$.