Yes, in the U.S., many credit cards offer CDW when renting a car as a benefit. It's much less common outside the U.S., so as an EU resident, you probably don't have it.
A sample reservation at Avis.com shows that CDW is $22.95 per day and ALI (Additional Liability Insurance) is $13.00 per day. There are of course taxes on top of that, but you can easily add one or both items at Avis.com to see the breakdown.
Here are the T&Cs from the Avis checkout page:
Optional Coverages
Loss Damage Waiver (LDW) is Optional on all car groups, *except car groups G, K, L, X and M (see details below).
Purchase reduces liability to zero in the event of an accident, fire, or theft to the Avis vehicle provided customer complies with the terms and conditions of the rental agreement. Theft Protection (TP) is included in the CDW.
*Exception: LDW is MANDATORY on 12 passenger vans, pickups and premium SUV's. Deductible on these car groups ranges between 800.00 USD to 1,000.00 USD.
If LDW is declined, and the rental incurs a loss due to damage or theft, the renter may be liable for up to the full replacement value of the vehicle.
Credit Card Policies
Avis Dominican Republic accepts Avis Charge Cards, American Express, Carte Blanche, Diners Club, Discover Card, Mastercard, Visa. An additional charge card hold will apply. The location will take 300.0 USD minimum as estimated rental charges as a Guarantee or Hold, depending on the type of vehicle and days rented. Note: Credit Cards without raised numbers are not accepted in the Dominican Republic.
If LDW is declined, the customer is liable for full value of the vehicle and must present a credit card at beginning of rental.
A minimum hold of 500.00 USD will be requested for the non-waivable excess, depending on car group rented in addition to the estimated rental charges. Customer is advised to contact the card issued to be sure about coverage prior to the rental. See LDW coverage information for more details.
Debit Card Policies
Debit cards are accepted, provided the customer purchases the Lost Damage Waiver (LDW).
Driver's License Requirements
Valid driver's license from country of residence and a passport must be presented in person at the time of rental. International driving license is required only for countries with non-Roman alphabet. International driving license is a translation only.
Drivers License
A valid driver's license from country of residence is required. Foreign renters must also present their valid passport in the driver's name. International driving permit (IDP) is required only for countries with non-Roman alphabet to serve as a translation.
Public Liability and Property Damage
Public liability insurance is in accordance with the insurance laws of the country. Customer may inquire at the time of rental for additional information of the benefits, conditions and acceptance.
Note: Some credit cards provide insurance coverage, with certain limitations, as a benefit of using the card to rent vehicles. The customer is advised to contact the card issuer before the rental. Any waiver must be discussed directly with the location upon arrival.
Additional Information
Drivers License:
A valid driving license from the renter's country of residence and a passport must be presented in person at the time of rental.
A local driving permit of $.95 USD per day will apply for all locations.
Here is similar information from Sabre (the system travel agents use):
Insurance
MOTOR LIABILITY INSURANCE - THIRD PARTY LIABILITY LIMITS
ARE INCLUDED IN RATES - . ONE PERSON - DOP
200,000.00 . TWO OR MORE PERSONS - DOP 400,000.00
. PROPERTY DAMAGE - DOP 200,000.00 ALI-ADDITIONAL
LIABILITY INSURANCE - 13.00 USD PER DAY PURCHASE OF ALI
PROVIDES 250,000 USD ADDITIONAL PROTECTION IN CASE OF BODILY
INJURY OR DEATH AND PROPERTY DAMAGE. CHECK AT COUNTER FOR
ADDITIONAL INFO.
Loss Damage Waiver
LDW - LOSS DAMAGE WAIVER 21.95 USD PER DAY - FOR ALL VEHICLES
LDW
IS OPTIONAL ON ALL CAR GROUPS *EXCEPT G,K,L,M,X -SEE
EXCEPTION
BELOW- PURCHASE REDUCES LIABILITY TO ZERO IN
THE EVENT
OF AN ACCIDENT, FIRE, OR THEFT TO THE AVIS VEHICLE
PROVIDED
CUSTOMER COMPLIES WITH THE TERMS AND CONDITIONS
OF THE
RENTAL AGREEMENT. THEFT PROTECTION IS INCLUDED
IN THE
LDW. IF LDW IS DECLINED, THE RENTAL INCURS A LOSS
DUE TO
DAMAGE OR THEFT THE RENTER MAY BE LIABLE FOR UP
TO THE
FULL REPLACEMENT VALUE OF THE VEHICLE. IF LDW IS
DECLINED,
THE RENTAL INCURS A LOSS DUE TO DAMAGE OR THEFT
THE RENTER
MAY BE LIABLE FOR UP TO THE FULL REPLACEMENT
VALUE OF THE
VEHICLE. *EXCEPTION- LDW IS MANDATORY ON 12
PASSENGER VANS,
PICKUPS AND PREMIUM SUVS. DEDUCTIBLE ON
THESE CAR GROUPS
RANGES BETWEEN 800.00 USD TO 1,000.00
USD. NOTE- THEFTS
THAT OCCUR DUE TO CUSTOMER NEGLIGENCE
-NOT LOCKING VEHICLE
OR SECURING KEYS- IS NOT COVERED.
IF ADDITIONAL PROTECTION
COVERAGE IS REQUIRED TO REDUCE
OR REMOVE THE INSURANCE
EXCESS, INQUIRE AT THE AVIS DESK
WHEN YOU COLLECT YOUR
CAR. IN ALL CASES, CUSTOMERS MUST
PRESENT A POLICE REPORT
FOR ANY INCIDENTS TO RECEIVE THE
COVERAGE. SOME CREDIT
CARDS PROVIDE INSURANCE COVERAGE,
WITH CERTAIN LIMITATIONS,
AS A BENFIT OF USING THE CARD
TO RENT VEHICLES. THE CUSTOMER
IS ADVISED TO CONTACT THE
CARD ISSUER BEFORE THE RENTAL
AND GET WRITTEN PROOF OF
THE COVERAGE TO BE PRESENTED AT
THE COUNTER AT THE TIME
OF RENTAL ON SPECIAL CATEGORY CAR
GROUPS, AS PICKUPS, LUXURY
AND PREMIUM SUVS, SPECIALTY
AND AND MINIVAN. ANY WAIVER
MUST BE DISCUSSED WITH THE
LOCATION UPON ARRIVAL. WRITTEN
PROOF SHOULD BE OBTAINED
FROM THE CREDIT CARD COMPANY ON
OFFICIAL LETTERHEAD NOTING
THEY WILL EXTEND LDW COVERAGE.
It looks like both CDW and ALI/LIS are both optional (on most vehicles), but that doesn't mean you won't be pressured into accepting one or both at the counter. Then again, if you have an accident, you won't be covered at all if you don't purchase the CDW and damage the vehicle, so depends on your risk profile.
As far as liability goes, some base level is included by law, but it's unclear what the law actually provides, and whether you feel this is enough to cover you.