July 24 (Reuters) - Sales of new U.S. single-family homes snapped a two-month skid in June, but higher mortgage rates and prices restrained the rebound and continue to sideline potential buyers from the housing market.

New home sales rose 1.6% to a seasonally adjusted annualized rate of 628,000 units last month from May's upwardly revised pace, the Commerce Department's Census Bureau said on Friday. New home sales, which are counted at the closing of a contract, account for a small share of U.S. home sales and tend to be volatile on a month-to-month basis. They fell 5.6% on a year-over-year basis in June.

Economists polled by Reuters had estimated the sales pace at 610,000 units.

The median new house price of $398,300 in June was 2.7% lower than a year earlier.

Steep borrowing costs are a key headwind for a housing market that remains on the back foot.

The average interest rate on a 30-year, fixed-rate mortgage - the most popular type of U.S. home loan - has in the last week risen to the highest level since last August with little prospect for an immediate break for would-be homebuyers, thanks to inflation-wariness among Federal Reserve officials and across bond markets.

Freddie Mac on Thursday said the average 30-year mortgage rate nationally had climbed this week to 6.58%. A day earlier the Mortgage Bankers Association said the 30-year mortgage contract rate hit 6.69% in the week ended July 17. Both rates were the highest in 11 months.

Mortgage rates have now climbed by around 0.60 percentage points since the U.S. and Israel launched attacks against Iran in late February, driving up global oil prices and helping fuel higher inflation more broadly.

Prices by the measure used by the Fed for its 2% inflation target are rising at roughly twice that pace, and bond markets are now convinced the U.S. central bank will respond soon with rate hikes. The Fed will hold a policy meeting next week, and while rate futures markets reflect only about a one-in-three chance of a hike at that meeting, the probability rises to nearly 100% at the following meeting in September.

Bond markets aren't waiting. Ten-year Treasury note yields, which serve as a pricing benchmark for 30-year mortgages, have risen by a quarter of a percentage point this month and are near their highest level in 18 months.