According to Catholic doctrine, is every woman that has an abortion automatically going to hell? This is something I've just heard but never verified. Does the Catholic church teach that every woman who has an abortion automatically earns hell no matter what? 

If so, then why? And how are other murderers in the bible granted eternal life--like David and Paul?
Au Contraire

There is no exception to the sin the Church is able to  absolve a penitent from1, even if they should sin until the last moment of their lives:


  Catechism of the Catholic Church | ARTICLE 10: "I BELIEVE IN THE
  FORGIVENESS OF SINS" | II. THE POWER OF THE KEYS,
  979
  In this battle against our inclination towards evil, who could be
  brave and watchful enough to escape every wound of sin? "If the Church
  has the power to forgive sins, then Baptism cannot be her only means
  of using the keys of the Kingdom of heaven received from Jesus Christ.
  The Church must be able to forgive all penitents their offenses, even if they should sin until the last moment of their lives."[524
  Roman Catechism I, 11,4.]




The only unforgivable sin is the sin against the Holy Spirit - matter for another answer.




  Mark 16:8-10
  (RSVCE)
  [...] Jesus Appears to Mary Magdalene   9 Now when he rose early
  on the first day of the week, he appeared first to Mary Mag′dalene,
  from whom he had cast out seven demons. 10 She went out and told
  those who had been with him, as they mourned and wept.


If you heard what you asked from one who has suffered from abortion, the above should be consoling and encouraging for them. The number 7 has a very significant meaning in Judaism. My take is that St. Mary Mag′dalene had sin in its totality [completely under the power of those 7 demons - inciting the seven deadly sins? - as far as God would allow them]. The LORD appeared to her first. First step is go, show yourself to the priest2 = confession [a good shepherd should be able to guide once there cf. MattGutting's answer esp. on excommunication, etc.]: “Take heart; rise, he is calling you.” [cf. Mk 10:49]




  Si 17:24
  (RSVCE)
  Yet to those who repent he grants a return,
  and he encourages those whose endurance is failing.




Rachel's Vineyard: This or the like may be of interest to women suffering after having had an abortion.



1. [cf. CCC 982]↩

2. [Matthew 8:4 (RSVCE)]↩