I found the following answer after posting it: $$\Delta=\epsilon\prod_c\frac{|G|}{|c|},\qquad\epsilon=(-1)^m,$$ where the product is taken over the conjugacy class. And $m$ is the number of pairs of complex conjugate irreducible characters.
Proof. On the one hand, the complex conjugate of the table is itself, up to $m$ transpositions of rows. This is because the conjugate of an IC is an IC. Therefore $$\overline{\det(TC)}=\epsilon\det(TC).$$ $\overline{\det(TC)}=\epsilon\det(TC)$$(TC stands for table of characters''.) Hence \det(TC) is real if m is even, pure imaginary if m is odd. hence \Delta is real and its sign is \epsilon. Now the characters form a unitary basis. Because a unitary matrix has a unit determinant, we may compute |\Delta| by taking any unitary basis. Take \phi_c(g) to be 0 if g\not\in c and |G|^{1/2}/|c|^{1/2} if g\in c. In particular |\Delta| is an integer because$$\frac{|G|}{|c|}=|{\mathcal Z}(a)|,\qquad a\in c.$$Another Proof: Let D be the diagonal matrix whose diagonal entries are the cardinals of the congugacy classes. We may assume that the first rows of TC are the real characters and the 2m last ones are the pairs of complex conjugate characters. Then the (i,j)-entry of M:=(TC)D(TC)^T is |G|\langle\overline{\chi_i},\chi_j\rangle. From the orthogonality relations, we see that M={\rm diag}(1,\ldots,1,J,\ldots,J) where$$J=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$$The number of blocks J is precisely m. Now take the determinant; we obtain \Delta\det D=(-1)^m|G|^r where r\times r is the size of TC. Hence the formula. 1 I found the following answer after posting it:$$\Delta=\epsilon\prod_c\frac{|G|}{|c|},\qquad\epsilon=(-1)^m,$$where the product is taken over the conjugacy class. And m is the number of pairs of complex conjugate irreducible characters. Proof. On the one hand, the complex conjugate of the table is itself, up to m transpositions of rows. This is because the conjugate of an IC is an IC. Therefore$$\overline{\det(TC)}=\epsilon\det(TC).$$Hence \det(TC) is real if m is even, pure imaginary if m is odd. hence \Delta is real and its sign is \epsilon. Now the characters form a unitary basis. Because a unitary matrix has a unit determinant, we may compute |\Delta| by taking any unitary basis. Take \phi_c(g) to be 0 if g\not\in c and |G|^{1/2}/|c|^{1/2} if g\in c. In particular |\Delta| is an integer because$$\frac{|G|}{|c|}=|{\mathcal Z}(a)|,\qquad a\in c.$\$