Why is the diode forward voltage constant?
The voltage across the diode does not remain at about 0.7 V. When you increase the current, the forward voltage also increases (here: 1N400x):



And when you increase the current even further, the power dissipation becomes too large, and the diode eventually becomes a LED (light-emitting diode) and shortly afterwards a SED (smoke-emitting diode). So a larger forward voltage cannot happen in practice.