I started working in February 20, 2017 and had an online W4 to fill out and I accidentally selected tax exempt. I am in the 28% tax bracket and live in Washington state. Upon realizing my mistake I have immediately updated my W4 but I owe a large amount of unpaid taxes for 2017 (I don't pay much attention to my direct deposits which is how it went unnoticed for so long). I have contacted the payroll at my company to see what they advise but it is the weekend and a large company so I don't know how long it will take for them to get back to me. What can I do to fix this as quickly as possible?
There's nothing your company can do to fix this. You have to pay the tax due directly to the IRS.
If you had little or no income for the previous year, 2016 (and thus little or no total tax), then you won't owe any penalties or interest as long as you pay the full tax due for 2017 by April 17, 2018. If your withholding for 2017 was less than your total tax for 2016 and you owe more than $1000, then you will owe interest for under-withholding. (It's called "Penalty for underpayment of estimated tax".) It's not a terribly large penalty. See Form 2210 and its instructions for more information. The IRS will calculate the penalty for you and send you a bill.
If you can't pay the tax due by April, then be sure to still file your return on time. You will owe penalties and interest as long as the tax remains unpaid. These penalties are much more severe than the penalty mentioned above. You should contact the IRS to establish a payment plan, which will help minimize the penalties--way better than simply not paying them.