Anakin is self-confident to a fault
The arrogance and pride that Anakin had - that Palpatine encouraged - was nearly limitless. He was about 22 years old and never planned ahead. He would wait until the last second to perform a task, wouldn't plan it out, and in his experience and according to his perspective everything always worked out well. He would even often ignore the advice or planning of his master in favor of his own last-minute decisions, and felt justified in doing so when things worked out.
It was this hubris that led him to take no action in the months, weeks, and days leading up to the birth event which he had foreseen was fatal to Padme.
The only thing he truly ever felt failure in was saving his mother's life, and he may have believed that if he had been there it would have been ok. This doesn't lead him to prepare in any special way for the impending birth, but to stick close to Padme, reluctant to leave her side without assurances from Palpatine that she would be watched over.
Anakin honestly believed that as long as he is near Padme, then no matter what happened he would be able to save her.
He didn't consider it a specialty, a process, or something he needed to prepare for. He assumed that he simply had enough power, and that he would learn anything additional he lacked on the fly during the event.
Perhaps Anakin could have saved her
We don't know what caused her death, not in terms that would help medical bots save her, anyway. However, there is evidence that as Darth Vader, Anakin had to use the force to maintain his own life and ability to move.
In a parallel universe, perhaps Anakin noticed her body was preparing for labor, and perhaps he stopped fighting with her and went to the medical facility, and perhaps he spent the rest of his life using the force to sustain her life. This one different choice would have him spending his force ability sustaining as much life in his wife as he did spend sustaining his own life as Vader.
This is, perhaps, something even the Jedi could have done, except for their creed preventing attachments. In fact, this creed points to the idea that if a Jedi used their power this way, where would they draw a line, assuming it's finite?