Padme didn't know because doing a medical examination might leak out and expose the father, not to mention she probably wanted to talk to Anakin about what to do next. Anakin was too concerned with his dreams and with Palpatine's machinations against the Jedi to pay attention to his wife and children.
Anakin Skywalker never knew Padmé was pregnant with twins, and he believed his child had died with his wife. He had no idea Padmé had given birth before she died, still less that the Jedi had spirited the children away and hidden them.
For example, a case could be made that no one discussed the pregnancy because Padmé hid the pregnancy with the assistance of her loyal handmaidens. It could also be argued that the father of her child was an open secret, and rather than assuming the father was Rush Clovis, everyone assumed it was Anakin Skywalker.
At some point after his transformation into the armored Darth Vader, Amidala's husband, the fallen Jedi Knight Anakin Skywalker came to visit the mausoleum, stricken by grief and remorse for his part in her death.
Leia's Force abilities and her lack of awareness of her true identity also contributed to Vader's inability to sense their familial connection. Vader's denial of Anakin Skywalker's legacy and his fear of getting hurt again prevented him from realizing the truth about his children until later in the original trilogy.
In order to keep the children safe from Vader, who was unaware of their births, it was decided that Luke would be taken to live with his uncle and aunt, Owen and Beru Lars, on Tatooine; meanwhile Leia was taken to Alderaan as the adopted child of Amidala's friend and colleague, Senator Bail Organa, and his wife, Queen ...
Vader discovers Leia is his daughter because of Luke.
Luke had only recently discovered the two were siblings and, even more recently, had revealed their connection to the princess. Vader likely sensed their emotional bond through the Force when speaking with his son.
Leia was not using the Force or was currently Force-sensitive during her confrontation with Vader. She began training as a Jedi years later. All these reasons prove my point: Vader didn't know Leia was his daughter, and felt no presence of the Force in her, as she was not currently strong with the Force.
In Star Wars Legends Leia was visited by the force ghost of her father Anakin Skywalker, the man who used to be Darth Vader. It wasn't a pleasant meeting as Leia was unable to forgive him for the things he'd done no matter how hard he begged, and it was the only time she saw him.
Anakin's reaction and joy upon hearing Padme's pregnancy is beautiful and heartwarming. He and Padme truly loved their children and would've been great parents if not for Palpatine's manipulations.
Honestly, I just couldn't figure out who she was or why she was even in the funeral procession. This also means that for the longest time I forgot I could just look up the credits of the film. Long story short, she is Padmé's niece, Pooja Naberrie, and was portrayed by Hayley Mooy.
The subtitles and script show some of Padme's final words to Obi Wan as “There is still good in him.” Every time I watch this I swear she says “Obi Wan... there's quite a name.” I have always thought of it as her recognizing him for everything he has done over the trilogy.
While in the Ewok village, Luke revealed his true lineage to Leia. She was shocked to discover they were brother and sister, though deep down, she always knew this to be true.
He eventually became so desperate to find a way to prevent Padmé from dying that his fear was ultimately the main reason that led to his fall to the dark side. Vader screaming for the loss of Padmé. Even as a Sith Lord, Anakin still loved Padmé very much, but felt great guilt for his actions against her.
Luke and Leia never sensed that they were twins until Return of the Jedi despite both being sensitive to the Force. Anakin Skywalker's twin children are the original Star Wars trilogy's driving force, having been separated after Padme's death.
Before he could enter, however, Vader was stopped by Sabé, Eirtaé, Rabé, Saché, and Dormé, former handmaidens of Amidala, who refused to let him enter into the grave so he would not desecrate it. However, Vader would not be denied. As he entered the tomb, he became flooded with memories of Amidala.
The first few slides of the comic book show a warrior attacking Vader, and when he dies he cries: "For Padme, and for Anakin." If this wasn't enough to get Vader feeling a little emotional, he then stumbled upon the tomb of Padme herself.
