וה' פקד את שרה כאשר אמר. בהריון. ויעש ה' לשרה כאשר דבר בלידה. וז"ש מיד ותהר ותלד, ונסמכה פרשה זו לפרשת כי עצור עצר ה' להודיע כי השי"ת הוא העוצר והוא הפוקד ואין זולתו. . וה' פקד את שרה כאשר אמר, “and G’d had remembered Sarah just as He had said.” This means she became pregnant.
ויעש ה' לשרה כאשר דבר, “and G’d did for Sarah as He had spoken.” This is a reference to her giving birth. This is why the Torah immediately follows by recording: “she became pregnant and gave birth.” The reason the Torah wrote this paragraph immediately after having informed us of the closing of the birth-canals in Avimelech’s household, etc., is to remind us that both the “so-called” natural process of becoming pregnant and giving birth and the inability to do so are in G’d’s hands. He alone is in charge of who becomes pregnant and who does not.
ובמדרש וה' פקד את שרה זשאה"כ (יחזקאל ה') וידעו כל עצי השדה כי אני ה' השפלתי עץ גבוה הגבהתי עץ שפל הובשתי עץ לח והפרחתי עץ יבש אני ה' דברתי ועשיתי, השפלתי עץ גבוה זה סנחריב שנאמר הנה אשור ארז בלבנון יפה קומה. הגבהתי עץ שפל אלו ישראל שמשפילין עצמן שנאמר (מלאכי ב׳:ט׳) וגם אני נתתי אתכם נבזים ושפלים. הובשתי עץ לח אבימלך שנא' כי עצור עצר ה', הפרחתי עץ יבש זו שרה. אני ה' דברתי ועשיתי. וה' פקד את שרה מהו פקד אמר רבי יצחק כתיב (במדבר ה׳:כ״ח) ואם לא נטמאה האשה וטהורה היא וגו' וזו נכנסה לביתו של פרעה ויצאה טהורה לביתו של אבימלך ויצאה טהורה, אינו בדין שתפקד, הוי אומר וה' פקד את שרה, ע"כ בב"ר. The Midrash (Tanchuma Vayera 15) uses this verse to remind us of Ezekiel 17,24: “Then shall all the trees of the field know that it is I the Lord who have abased the lofty tree and exalted the lowly tree, who have dried up the green tree and made the withered tree bud. I the Lord have spoken, and I will act.” According to this Midrash the prophet referred to Sancheriv when he spoke about G’d “abasing the lofty tree,” seeing that in Ezekiel 31,3 Ashur (Sancheriv’s country) is described as “a cedar in Lebanon with beautiful branches of lofty stature.” On the other hand, the words: “I have exalted the lowly tree,” are a reference to the Jewish people who debased themselves as we know from Maleachi 2,9: “I too have made you despicable and vile in the eyes of all the people...said the Lord of Hosts.” The words: “I have dried out the green tree,” are a reference to Avimelech of whom the Torah said that G’d had closed up all his orifices, whereas the words “and I have made the withered tree bud,” refer to Sarah. The words: “I the Lord have spoken and I have acted,” refer to our verse in which G’d is reported as having remembered Sarah.
As to the choice of the word והשם, “and the Lord,” Rabbi Yitzchak said: we find that the Torah writes in connection with the Sotah, the wife suspected of infidelity who had secreted herself with another man (Numbers 5,28), “but if the woman had not defiled herself and remained pure, etc.” Such a description fits Sarah who had been alone with Pharaoh in Egypt and Avimelech in the land of the Philistines and had retained her purity on both occasions. Such a woman had certainly established a claim that G’d should remember her. This is why the paragraph starts with the prefix ו, i.e. “and” in front of the word השם, “The Lord.”