The answer is Yes, here is the statute, Code of Civil Procedure § 377.34:

377.34. In an action or proceeding by a decedent's personal
representative or successor in interest on the decedent's cause of
action, the damages recoverable are limited to the loss or damage
that the decedent sustained or incurred before death, including any
penalties or punitive or exemplary damages that the decedent would
have been entitled to recover had the decedent lived, and do not
include damages for pain, suffering, or disfigurement.
From
http://leginfo.ca.gov/cgi-bin/displaycode?section=ccp&group=00001-01000&file=377.30-377.35

The phrase "incurred before death" includes unpaid medical bills.

Here are some other relevant statutes:

377.30. A cause of action that survives the death of the person
entitled to commence an action or proceeding passes to the decedent's
successor in interest, subject to Chapter 1 (commencing with Section
7000) of Part 1 of Division 7 of the Probate Code, and an action may
be commenced by the decedent's personal representative or, if none,
by the decedent's successor in interest.

377.31. On motion after the death of a person who commenced an
action or proceeding, the court shall allow a pending action or
proceeding that does not abate to be continued by the decedent's
personal representative or, if none, by the decedent's successor in
interest.

377.20. (a) Except as otherwise provided by statute, a cause of
action for or against a person is not lost by reason of the person's
death, but survives subject to the applicable limitations period.
(b) This section applies even though a loss or damage occurs
simultaneously with or after the death of a person who would have
been liable if the person's death had not preceded or occurred
simultaneously with the loss or damage.

377.21. A pending action or proceeding does not abate by the death
of a party if the cause of action survives.
Via
http://leginfo.ca.gov/cgi-bin/calawquery?codesection=ccp&codebody=&hits=20

I hope this information is helpful.